{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0001","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67643bf542456963a2b1fe7f57d808a8ed215b275a9a2e672c701094ad8da0e1","family_id":"qf-mod31-quotring-classes","seed":202610314001,"semantic_template":"qf-mod31-quotring-classes/{\"d\":2,\"n\":4}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_4 with residues 0 through 3 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 2], [[0, 2], [1, 3]], 2, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 2], [[0, 2], [1, 3]], 2, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0002","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e3a8c1f8f33501f067e37a5576dc548c7642bf5561d80af4fc0847e5fb0dfcb","family_id":"qf-mod31-quotring-classes","seed":202610314002,"semantic_template":"qf-mod31-quotring-classes/{\"d\":3,\"n\":6}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 3], [[0, 3], [1, 4], [2, 5]], 3, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 3], [[0, 3], [1, 4], [2, 5]], 3, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0003","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71b92d82a11c310f61af6c82de3202862a90e520883bbfd908446a2e3a6e6777","family_id":"qf-mod31-quotring-classes","seed":202610314003,"semantic_template":"qf-mod31-quotring-classes/{\"d\":4,\"n\":8}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], 4, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], 4, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0004","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba12c3cec274a9fc2882a211f68cc1ff9d13683a38893cf00140b8ddbf6d852b","family_id":"qf-mod31-quotring-classes","seed":202610314004,"semantic_template":"qf-mod31-quotring-classes/{\"d\":5,\"n\":10}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 5], [[0, 5], [1, 6], [2, 7], [3, 8], [4, 9]], 5, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 5], [[0, 5], [1, 6], [2, 7], [3, 8], [4, 9]], 5, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0005","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f69a02485005503358c839b77d922fcbd12aba0fcbe2f1778002515c25a8682b","family_id":"qf-mod31-quotring-classes","seed":202610314005,"semantic_template":"qf-mod31-quotring-classes/{\"d\":6,\"n\":12}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 6], [[0, 6], [1, 7], [2, 8], [3, 9], [4, 10], [5, 11]], 6, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 6], [[0, 6], [1, 7], [2, 8], [3, 9], [4, 10], [5, 11]], 6, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0006","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cad64a8a29317f788f66f3baf5c5cfdfe33ed0a8a674cc78573b81293170f950","family_id":"qf-mod31-quotring-classes","seed":202610314006,"semantic_template":"qf-mod31-quotring-classes/{\"d\":2,\"n\":6}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 2, 4], [[0, 2, 4], [1, 3, 5]], 2, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 2, 4], [[0, 2, 4], [1, 3, 5]], 2, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0007","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ce6feaba03c2e52a8148b03ef1740a7d2a7189c9359ce06d61ccc18a984a1a0","family_id":"qf-mod31-quotring-classes","seed":202610314007,"semantic_template":"qf-mod31-quotring-classes/{\"d\":3,\"n\":9}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_9 with residues 0 through 8 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 3, 6], [[0, 3, 6], [1, 4, 7], [2, 5, 8]], 3, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 3, 6], [[0, 3, 6], [1, 4, 7], [2, 5, 8]], 3, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0008","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e35b3f47779d1c8d9fc10f4b9648d3be108414231119e0c6a322bd9fd0373b30","family_id":"qf-mod31-quotring-classes","seed":202610314008,"semantic_template":"qf-mod31-quotring-classes/{\"d\":4,\"n\":12}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 4, 8], [[0, 4, 8], [1, 5, 9], [2, 6, 10], [3, 7, 11]], 4, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 4, 8], [[0, 4, 8], [1, 5, 9], [2, 6, 10], [3, 7, 11]], 4, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0009","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45df9a59cf913ff6fcd4e2ab23b49d240fd95d7f2b8eefd0be6b7defc7e9e352","family_id":"qf-mod31-quotring-classes","seed":202610314009,"semantic_template":"qf-mod31-quotring-classes/{\"d\":5,\"n\":15}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 5, 10], [[0, 5, 10], [1, 6, 11], [2, 7, 12], [3, 8, 13], [4, 9, 14]], 5, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 5, 10], [[0, 5, 10], [1, 6, 11], [2, 7, 12], [3, 8, 13], [4, 9, 14]], 5, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0010","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14f15866d6cf8a71173e60d7e59d430186e3a4640712c5f18ec45fef4c35803b","family_id":"qf-mod31-quotring-classes","seed":202610314010,"semantic_template":"qf-mod31-quotring-classes/{\"d\":6,\"n\":18}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 6, 12], [[0, 6, 12], [1, 7, 13], [2, 8, 14], [3, 9, 15], [4, 10, 16], [5, 11, 17]], 6, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 6, 12], [[0, 6, 12], [1, 7, 13], [2, 8, 14], [3, 9, 15], [4, 10, 16], [5, 11, 17]], 6, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0011","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc2a42e8f19813dc01267e6b929aaa53f6f5ae66a523181dd76471df6009a344","family_id":"qf-mod31-quotring-classes","seed":202610314011,"semantic_template":"qf-mod31-quotring-classes/{\"d\":2,\"n\":8}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 2, 4, 6], [[0, 2, 4, 6], [1, 3, 5, 7]], 2, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 2, 4, 6], [[0, 2, 4, 6], [1, 3, 5, 7]], 2, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0012","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5bf97539bb25eb6ebd4021097b6c619024e8eb99b144466044ca168c42f2ae3","family_id":"qf-mod31-quotring-classes","seed":202610314012,"semantic_template":"qf-mod31-quotring-classes/{\"d\":3,\"n\":12}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 3, 6, 9], [[0, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]], 3, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 3, 6, 9], [[0, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]], 3, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0013","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16b282a425439d33b350df085129a1833c05e161f1551436a3dc24af8bc41379","family_id":"qf-mod31-quotring-classes","seed":202610314013,"semantic_template":"qf-mod31-quotring-classes/{\"d\":4,\"n\":16}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_16 with residues 0 through 15 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 4, 8, 12], [[0, 4, 8, 12], [1, 5, 9, 13], [2, 6, 10, 14], [3, 7, 11, 15]], 4, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 4, 8, 12], [[0, 4, 8, 12], [1, 5, 9, 13], [2, 6, 10, 14], [3, 7, 11, 15]], 4, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0014","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3119f630a655a9da79e9d895ec322d9f9d7fbda391be4a920805b9b6f454b809","family_id":"qf-mod31-quotring-classes","seed":202610314014,"semantic_template":"qf-mod31-quotring-classes/{\"d\":5,\"n\":20}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 5, 10, 15], [[0, 5, 10, 15], [1, 6, 11, 16], [2, 7, 12, 17], [3, 8, 13, 18], [4, 9, 14, 19]], 5, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 5, 10, 15], [[0, 5, 10, 15], [1, 6, 11, 16], [2, 7, 12, 17], [3, 8, 13, 18], [4, 9, 14, 19]], 5, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0015","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3d17943f9a021db0f628d911c2f87684893fb6e5aa02c29350c899f0e2f7ab9","family_id":"qf-mod31-quotring-classes","seed":202610314015,"semantic_template":"qf-mod31-quotring-classes/{\"d\":6,\"n\":24}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 6, 12, 18], [[0, 6, 12, 18], [1, 7, 13, 19], [2, 8, 14, 20], [3, 9, 15, 21], [4, 10, 16, 22], [5, 11, 17, 23]], 6, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 6, 12, 18], [[0, 6, 12, 18], [1, 7, 13, 19], [2, 8, 14, 20], [3, 9, 15, 21], [4, 10, 16, 22], [5, 11, 17, 23]], 6, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0016","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a74dafd99052e39c200ee01a2ad1b2a049b6065284ad4d780d535531439b69a","family_id":"qf-mod31-quotring-classes","seed":202610314016,"semantic_template":"qf-mod31-quotring-classes/{\"d\":2,\"n\":10}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8], [[0, 2, 4, 6, 8], [1, 3, 5, 7, 9]], 2, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8], [[0, 2, 4, 6, 8], [1, 3, 5, 7, 9]], 2, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0017","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb37d19b46e3cecd652a6b7596dc9d344e424a2ac5e33d042d06f2a103faca87","family_id":"qf-mod31-quotring-classes","seed":202610314017,"semantic_template":"qf-mod31-quotring-classes/{\"d\":3,\"n\":15}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 3, 6, 9, 12], [[0, 3, 6, 9, 12], [1, 4, 7, 10, 13], [2, 5, 8, 11, 14]], 3, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 3, 6, 9, 12], [[0, 3, 6, 9, 12], [1, 4, 7, 10, 13], [2, 5, 8, 11, 14]], 3, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0018","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ce5a66e1796046dc93dfd726c516a689452865d83e39f66ff98b471dac4e304","family_id":"qf-mod31-quotring-classes","seed":202610314018,"semantic_template":"qf-mod31-quotring-classes/{\"d\":4,\"n\":20}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 4, 8, 12, 16], [[0, 4, 8, 12, 16], [1, 5, 9, 13, 17], [2, 6, 10, 14, 18], [3, 7, 11, 15, 19]], 4, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 4, 8, 12, 16], [[0, 4, 8, 12, 16], [1, 5, 9, 13, 17], [2, 6, 10, 14, 18], [3, 7, 11, 15, 19]], 4, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0019","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76b949c70e351fd4fe947dc63fe09af13b8f3c315cfabc2fa89cc2d2bfb2ac67","family_id":"qf-mod31-quotring-classes","seed":202610314019,"semantic_template":"qf-mod31-quotring-classes/{\"d\":5,\"n\":25}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_25 with residues 0 through 24 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 5, 10, 15, 20], [[0, 5, 10, 15, 20], [1, 6, 11, 16, 21], [2, 7, 12, 17, 22], [3, 8, 13, 18, 23], [4, 9, 14, 19, 24]], 5, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 5, 10, 15, 20], [[0, 5, 10, 15, 20], [1, 6, 11, 16, 21], [2, 7, 12, 17, 22], [3, 8, 13, 18, 23], [4, 9, 14, 19, 24]], 5, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0020","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50c354b7cca06d5321215cc6448db1a8da4715ecaa223f694036868970772c9e","family_id":"qf-mod31-quotring-classes","seed":202610314020,"semantic_template":"qf-mod31-quotring-classes/{\"d\":6,\"n\":30}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 6, 12, 18, 24], [[0, 6, 12, 18, 24], [1, 7, 13, 19, 25], [2, 8, 14, 20, 26], [3, 9, 15, 21, 27], [4, 10, 16, 22, 28], [5, 11, 17, 23, 29]], 6, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 6, 12, 18, 24], [[0, 6, 12, 18, 24], [1, 7, 13, 19, 25], [2, 8, 14, 20, 26], [3, 9, 15, 21, 27], [4, 10, 16, 22, 28], [5, 11, 17, 23, 29]], 6, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0021","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6688cf385a3b0f7c28c4ef9c29820c0aee836499121ed166640f58b032aafa5","family_id":"qf-mod31-quotring-classes","seed":202610314021,"semantic_template":"qf-mod31-quotring-classes/{\"d\":2,\"n\":12}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10], [[0, 2, 4, 6, 8, 10], [1, 3, 5, 7, 9, 11]], 2, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10], [[0, 2, 4, 6, 8, 10], [1, 3, 5, 7, 9, 11]], 2, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0022","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db0d07c1cf03e67a37e1a20013df57e52c86acb2f1fb6ba0e421d75dce618c6e","family_id":"qf-mod31-quotring-classes","seed":202610314022,"semantic_template":"qf-mod31-quotring-classes/{\"d\":3,\"n\":18}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 3, 6, 9, 12, 15], [[0, 3, 6, 9, 12, 15], [1, 4, 7, 10, 13, 16], [2, 5, 8, 11, 14, 17]], 3, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 3, 6, 9, 12, 15], [[0, 3, 6, 9, 12, 15], [1, 4, 7, 10, 13, 16], [2, 5, 8, 11, 14, 17]], 3, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0023","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"983797d7539ddd2f7b6a1936f6e6ed6eef6ee0cb094f09ac763dbcc7aae7c933","family_id":"qf-mod31-quotring-classes","seed":202610314023,"semantic_template":"qf-mod31-quotring-classes/{\"d\":4,\"n\":24}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 4, 8, 12, 16, 20], [[0, 4, 8, 12, 16, 20], [1, 5, 9, 13, 17, 21], [2, 6, 10, 14, 18, 22], [3, 7, 11, 15, 19, 23]], 4, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 4, 8, 12, 16, 20], [[0, 4, 8, 12, 16, 20], [1, 5, 9, 13, 17, 21], [2, 6, 10, 14, 18, 22], [3, 7, 11, 15, 19, 23]], 4, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0024","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49694d4ae3e6a322cc54b5eeddbaf55717094d57c67565fa86fdb9b2ef5bf62e","family_id":"qf-mod31-quotring-classes","seed":202610314024,"semantic_template":"qf-mod31-quotring-classes/{\"d\":5,\"n\":30}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 5, 10, 15, 20, 25], [[0, 5, 10, 15, 20, 25], [1, 6, 11, 16, 21, 26], [2, 7, 12, 17, 22, 27], [3, 8, 13, 18, 23, 28], [4, 9, 14, 19, 24, 29]], 5, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 5, 10, 15, 20, 25], [[0, 5, 10, 15, 20, 25], [1, 6, 11, 16, 21, 26], [2, 7, 12, 17, 22, 27], [3, 8, 13, 18, 23, 28], [4, 9, 14, 19, 24, 29]], 5, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0025","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d50b2f8cc2fbda9710c54e528153c735248c05489392f398891a8002ee241704","family_id":"qf-mod31-quotring-classes","seed":202610314025,"semantic_template":"qf-mod31-quotring-classes/{\"d\":6,\"n\":36}"},"fields":{"title":"Build the Quotient from Full Additive Cosets","stem":"Use the parent ring Z_36 with residues 0 through 35 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).","answer":{"kind":"expression","value":"([0, 6, 12, 18, 24, 30], [[0, 6, 12, 18, 24, 30], [1, 7, 13, 19, 25, 31], [2, 8, 14, 20, 26, 32], [3, 9, 15, 21, 27, 33], [4, 10, 16, 22, 28, 34], [5, 11, 17, 23, 29, 35]], 6, proper_two_sided_ideal_in_commutative_Zn)","explanation":"List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 6, 12, 18, 24, 30], [[0, 6, 12, 18, 24, 30], [1, 7, 13, 19, 25, 31], [2, 8, 14, 20, 26, 32], [3, 9, 15, 21, 27, 33], [4, 10, 16, 22, 28, 34], [5, 11, 17, 23, 29, 35]], 6, proper_two_sided_ideal_in_commutative_Zn)."},"skills":["Build the Quotient from Full Additive Cosets"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0026","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c6843147e0f0dd8413683b7f2c18cea26d896c66d41e15f2e10650c8dc33915","family_id":"qf-mod31-quotring-add","seed":202610314026,"semantic_template":"qf-mod31-quotring-add/{\"a\":0,\"b\":1,\"d\":2,\"n\":4}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_4 with residues 0 through 3 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=0, b=1. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(0, 1, 1, 1, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 1, 1, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0027","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4a91fd07ab996662ac314bfefc791f108e87fb4a5e112f2a288aa443b2c99c5","family_id":"qf-mod31-quotring-add","seed":202610314027,"semantic_template":"qf-mod31-quotring-add/{\"a\":1,\"b\":3,\"d\":3,\"n\":6}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=1, b=3. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(1, 0, 4, 1, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (1, 0, 4, 1, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0028","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21c98e2c48220de3da17282e9cfb484c763676e01c508b91ff282a517018fd8b","family_id":"qf-mod31-quotring-add","seed":202610314028,"semantic_template":"qf-mod31-quotring-add/{\"a\":2,\"b\":5,\"d\":4,\"n\":8}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=2, b=5. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(2, 1, 7, 3, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (2, 1, 7, 3, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0029","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ff8fb6f8c5f391d82148c605477a6d7d226f5b5cdfb77b80db5bd690144bb2a","family_id":"qf-mod31-quotring-add","seed":202610314029,"semantic_template":"qf-mod31-quotring-add/{\"a\":3,\"b\":7,\"d\":5,\"n\":10}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=3, b=7. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(3, 2, 0, 0, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 2, 0, 0, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0030","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74d82e9a3ab37acc47caab83be73bd64d0012f81069d3c1da3555c375f37eba3","family_id":"qf-mod31-quotring-add","seed":202610314030,"semantic_template":"qf-mod31-quotring-add/{\"a\":4,\"b\":9,\"d\":6,\"n\":12}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=4, b=9. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(4, 3, 1, 1, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (4, 3, 1, 1, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0031","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"deac090863503bd215c6a517de6767f0e372c407be4f0218bebfca42ec91784b","family_id":"qf-mod31-quotring-add","seed":202610314031,"semantic_template":"qf-mod31-quotring-add/{\"a\":5,\"b\":5,\"d\":2,\"n\":6}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=5, b=5. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(1, 1, 4, 0, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (1, 1, 4, 0, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0032","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7044b251e253f5aab5e2bffe72e66a0d01f263c7188bc12aee6e4cc03e0753cc","family_id":"qf-mod31-quotring-add","seed":202610314032,"semantic_template":"qf-mod31-quotring-add/{\"a\":6,\"b\":4,\"d\":3,\"n\":9}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_9 with residues 0 through 8 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=6, b=4. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(0, 1, 1, 1, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 1, 1, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0033","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e61390fe7fff7c3c9f3ff08f888470acc326de93edd18afe1b1f89eee4c41ee4","family_id":"qf-mod31-quotring-add","seed":202610314033,"semantic_template":"qf-mod31-quotring-add/{\"a\":7,\"b\":3,\"d\":4,\"n\":12}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=7, b=3. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(3, 3, 10, 2, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 3, 10, 2, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0034","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"068446a2bf2a259c482787a68c5338d8a119b411f64f28276f68201b9ce49df9","family_id":"qf-mod31-quotring-add","seed":202610314034,"semantic_template":"qf-mod31-quotring-add/{\"a\":8,\"b\":2,\"d\":5,\"n\":15}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=8, b=2. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(3, 2, 10, 0, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 2, 10, 0, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0035","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d011cbc06cb22a4a8cbae048ac10011dd16d7286fdf7891cbc83aaa545778b7","family_id":"qf-mod31-quotring-add","seed":202610314035,"semantic_template":"qf-mod31-quotring-add/{\"a\":9,\"b\":1,\"d\":6,\"n\":18}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=9, b=1. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(3, 1, 10, 4, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 1, 10, 4, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0036","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ce6dcaa34abc0873117bb6e7953bc7897a5311a9b16860ecff7db772b660b10","family_id":"qf-mod31-quotring-add","seed":202610314036,"semantic_template":"qf-mod31-quotring-add/{\"a\":2,\"b\":5,\"d\":2,\"n\":8}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=2, b=5. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(0, 1, 7, 1, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 7, 1, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0037","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d287eb324ff5960f4d3107b145f14763db046a01632b9c01379731059e61d82","family_id":"qf-mod31-quotring-add","seed":202610314037,"semantic_template":"qf-mod31-quotring-add/{\"a\":11,\"b\":11,\"d\":3,\"n\":12}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=11, b=11. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(2, 2, 10, 1, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (2, 2, 10, 1, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0038","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c06d6929cfbbd3057496967f78c7bb82035930d6adbb8fd04f035db2595c87c8","family_id":"qf-mod31-quotring-add","seed":202610314038,"semantic_template":"qf-mod31-quotring-add/{\"a\":12,\"b\":9,\"d\":4,\"n\":16}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_16 with residues 0 through 15 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=12, b=9. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(0, 1, 5, 1, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 5, 1, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0039","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5824af57ac0d6d23b6a8f10ec45ca06478469e834c0dfc058cc72b7f40afe531","family_id":"qf-mod31-quotring-add","seed":202610314039,"semantic_template":"qf-mod31-quotring-add/{\"a\":13,\"b\":7,\"d\":5,\"n\":20}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=13, b=7. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(3, 2, 0, 0, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 2, 0, 0, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0040","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83c9b77940eb61ed86986a6c9491264fece11ae3dad6b4e7fb588a27af47a8d6","family_id":"qf-mod31-quotring-add","seed":202610314040,"semantic_template":"qf-mod31-quotring-add/{\"a\":14,\"b\":5,\"d\":6,\"n\":24}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=14, b=5. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(2, 5, 19, 1, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (2, 5, 19, 1, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0041","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7add899e141c5a7d2bbcf04ad49b93dd5730279e3f834694798885f1ae73acda","family_id":"qf-mod31-quotring-add","seed":202610314041,"semantic_template":"qf-mod31-quotring-add/{\"a\":5,\"b\":1,\"d\":2,\"n\":10}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=5, b=1. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(1, 1, 6, 0, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (1, 1, 6, 0, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0042","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac84b500249fb1db37a8847ae64c8b27940ad3c193221163f7d456be03d29c0a","family_id":"qf-mod31-quotring-add","seed":202610314042,"semantic_template":"qf-mod31-quotring-add/{\"a\":1,\"b\":3,\"d\":3,\"n\":15}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=1, b=3. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(1, 0, 4, 1, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (1, 0, 4, 1, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0043","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65fdf7a71916c6499910b999bef49356d90beb88ad4e94f00c9c19f903ff6049","family_id":"qf-mod31-quotring-add","seed":202610314043,"semantic_template":"qf-mod31-quotring-add/{\"a\":17,\"b\":15,\"d\":4,\"n\":20}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=17, b=15. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(1, 3, 12, 0, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (1, 3, 12, 0, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0044","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75d4d37be9f1679e8382710a4cac1895be138c500e692b04c8ad7085606b46ec","family_id":"qf-mod31-quotring-add","seed":202610314044,"semantic_template":"qf-mod31-quotring-add/{\"a\":18,\"b\":12,\"d\":5,\"n\":25}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_25 with residues 0 through 24 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=18, b=12. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(3, 2, 5, 0, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 2, 5, 0, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0045","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5fa0b8dafee12555f8c3d97cb01cbe138b64d217894aa645a66f56562e00312","family_id":"qf-mod31-quotring-add","seed":202610314045,"semantic_template":"qf-mod31-quotring-add/{\"a\":19,\"b\":9,\"d\":6,\"n\":30}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=19, b=9. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(1, 3, 28, 4, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (1, 3, 28, 4, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0046","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab73ffe871052e32b39e7a69b5ad5fa5739c758777d67589d25fc0cc8c247e4d","family_id":"qf-mod31-quotring-add","seed":202610314046,"semantic_template":"qf-mod31-quotring-add/{\"a\":8,\"b\":5,\"d\":2,\"n\":12}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=8, b=5. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(0, 1, 1, 1, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 1, 1, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0047","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fec80ee3333e8462649689217720eefd663f97670998b4586d269fbc18e9fd0","family_id":"qf-mod31-quotring-add","seed":202610314047,"semantic_template":"qf-mod31-quotring-add/{\"a\":3,\"b\":7,\"d\":3,\"n\":18}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=3, b=7. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(0, 1, 10, 1, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 10, 1, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0048","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7f110d9a2ec5feee5621d75e671687f7e8c6fa82e6eed8601070a5d147676d1","family_id":"qf-mod31-quotring-add","seed":202610314048,"semantic_template":"qf-mod31-quotring-add/{\"a\":22,\"b\":21,\"d\":4,\"n\":24}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=22, b=21. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(2, 1, 19, 3, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (2, 1, 19, 3, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0049","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f62f2fd1104b690ac781583d132a39bedfdf11957eaaba3dd51c4bad0340e20","family_id":"qf-mod31-quotring-add","seed":202610314049,"semantic_template":"qf-mod31-quotring-add/{\"a\":23,\"b\":17,\"d\":5,\"n\":30}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=23, b=17. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(3, 2, 10, 0, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 2, 10, 0, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0050","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a75050c123ff5b0f52031fe7cfe22fdc1ba591f24feb636205dfb48b89959ce","family_id":"qf-mod31-quotring-add","seed":202610314050,"semantic_template":"qf-mod31-quotring-add/{\"a\":24,\"b\":13,\"d\":6,\"n\":36}"},"fields":{"title":"Add in the Parent Ring, Then Take the Quotient Class","stem":"Use the parent ring Z_36 with residues 0 through 35 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=24, b=13. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).","answer":{"kind":"expression","value":"(0, 1, 1, 1, [a]+[b]=[a+b])","explanation":"Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 1, 1, [a]+[b]=[a+b])."},"skills":["Add in the Parent Ring, Then Take the Quotient Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0051","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3ee16267a1867b7d3e55a3448fa0f33feb2f337ed8370249cf33757f1225edc","family_id":"qf-mod31-quotring-multiply","seed":202610314051,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":0,\"b\":1,\"d\":2,\"n\":4}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_4 with residues 0 through 3 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=0, b=1. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(0, 1, 0, 0, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 0, 0, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0052","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59c9e7039ff3d0113c802db72c3018c539f58276f0ba0b0a103626102ccb7d4e","family_id":"qf-mod31-quotring-multiply","seed":202610314052,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":1,\"b\":3,\"d\":3,\"n\":6}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=1, b=3. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(1, 0, 3, 0, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (1, 0, 3, 0, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0053","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ff7d3535628a52803a2219580771867498378b37b1cc8bbad03e0264b5ddb85","family_id":"qf-mod31-quotring-multiply","seed":202610314053,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":2,\"b\":5,\"d\":4,\"n\":8}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=2, b=5. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(2, 1, 2, 2, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (2, 1, 2, 2, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0054","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a02554c3c9809f2a7065812b0c48ebabeb5626ef978e3e603187b4307782bc29","family_id":"qf-mod31-quotring-multiply","seed":202610314054,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":3,\"b\":7,\"d\":5,\"n\":10}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=3, b=7. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(3, 2, 1, 1, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 2, 1, 1, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0055","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54de43db074af30abd72930e2d460186f66e9cbda36384565f5fb9d973028b16","family_id":"qf-mod31-quotring-multiply","seed":202610314055,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":4,\"b\":9,\"d\":6,\"n\":12}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=4, b=9. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(4, 3, 0, 0, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (4, 3, 0, 0, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0056","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93ab49f0147557206fc9b4afcbe456db0d9fab71ce6b43da31d3bb6919775589","family_id":"qf-mod31-quotring-multiply","seed":202610314056,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":5,\"b\":5,\"d\":2,\"n\":6}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=5, b=5. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(1, 1, 1, 1, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (1, 1, 1, 1, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0057","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0de441d328896bfc486b4296b2b7afaf10b7b967e027750506828581342186d2","family_id":"qf-mod31-quotring-multiply","seed":202610314057,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":6,\"b\":4,\"d\":3,\"n\":9}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_9 with residues 0 through 8 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=6, b=4. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(0, 1, 6, 0, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 6, 0, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0058","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65bef0c0b5b46d578b364f29896955d53b361b44bb6788b1ba6695e608ad064d","family_id":"qf-mod31-quotring-multiply","seed":202610314058,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":7,\"b\":3,\"d\":4,\"n\":12}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=7, b=3. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(3, 3, 9, 1, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 3, 9, 1, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0059","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0c91bb7529733f9198e99f3e0d547a0c31d6fe819e5e811f9eb6d40ec78b2a7","family_id":"qf-mod31-quotring-multiply","seed":202610314059,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":8,\"b\":2,\"d\":5,\"n\":15}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=8, b=2. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(3, 2, 1, 1, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 2, 1, 1, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0060","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4f6bd8b659423a8207ac0b8d2153a3aa55bfca5fc164e21fe48d516a198c191","family_id":"qf-mod31-quotring-multiply","seed":202610314060,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":9,\"b\":1,\"d\":6,\"n\":18}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=9, b=1. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(3, 1, 9, 3, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 1, 9, 3, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0061","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0339d1550923811cc2bc5fa94737399c37877bece50b1366c6ce0bed375e0479","family_id":"qf-mod31-quotring-multiply","seed":202610314061,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":2,\"b\":5,\"d\":2,\"n\":8}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=2, b=5. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(0, 1, 2, 0, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 2, 0, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0062","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"245d0276cb3437f57d994ca4a3b0107d1acb601cb9016c10dd132fdeb8935438","family_id":"qf-mod31-quotring-multiply","seed":202610314062,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":11,\"b\":11,\"d\":3,\"n\":12}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=11, b=11. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(2, 2, 1, 1, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (2, 2, 1, 1, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0063","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d340cd64fc0ffa969046b65fb0630005a0b12800f12063e6cca3506f0c825e8","family_id":"qf-mod31-quotring-multiply","seed":202610314063,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":12,\"b\":9,\"d\":4,\"n\":16}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_16 with residues 0 through 15 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=12, b=9. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(0, 1, 12, 0, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 12, 0, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0064","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e767de2bf1985f9a9b44bb39c1506a9036a006db01bb03c7980e60bdc3c3cb5b","family_id":"qf-mod31-quotring-multiply","seed":202610314064,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":13,\"b\":7,\"d\":5,\"n\":20}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=13, b=7. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(3, 2, 11, 1, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 2, 11, 1, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0065","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4031312fdaf1aba459155cae811e87ddfdff3cf7144f349c015cd9131afa11c3","family_id":"qf-mod31-quotring-multiply","seed":202610314065,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":14,\"b\":5,\"d\":6,\"n\":24}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=14, b=5. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(2, 5, 22, 4, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (2, 5, 22, 4, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0066","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"470cd2a38f16f8de5e0a6107f86a27c2c7c9b664b38cf8d26eb75f46dcd2dc3c","family_id":"qf-mod31-quotring-multiply","seed":202610314066,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":5,\"b\":1,\"d\":2,\"n\":10}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=5, b=1. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(1, 1, 5, 1, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (1, 1, 5, 1, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0067","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2706a03e1e39333cecf9214b174fb924fc1ac35ad450dbc043f7d133c20e96b0","family_id":"qf-mod31-quotring-multiply","seed":202610314067,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":1,\"b\":3,\"d\":3,\"n\":15}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=1, b=3. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(1, 0, 3, 0, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (1, 0, 3, 0, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0068","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17e15f3059cb4b8e5278f68ed1d855fe1e3e7c82c681b4d23b07844082fde7cb","family_id":"qf-mod31-quotring-multiply","seed":202610314068,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":17,\"b\":15,\"d\":4,\"n\":20}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=17, b=15. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(1, 3, 15, 3, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (1, 3, 15, 3, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0069","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0494c786b568b64e34fd40468390f6ef81b4fef7fec7663553dcda5ed80cd217","family_id":"qf-mod31-quotring-multiply","seed":202610314069,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":18,\"b\":12,\"d\":5,\"n\":25}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_25 with residues 0 through 24 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=18, b=12. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(3, 2, 16, 1, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 2, 16, 1, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0070","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e684506a9b4733b2fc66f77f451eab8d382974e2fc14d96e49edf1ded500e478","family_id":"qf-mod31-quotring-multiply","seed":202610314070,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":19,\"b\":9,\"d\":6,\"n\":30}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=19, b=9. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(1, 3, 21, 3, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (1, 3, 21, 3, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0071","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"897f8bcf2c1c565647141c24b2e6dc0fbc1316416f55fbc78afa167a70c914c2","family_id":"qf-mod31-quotring-multiply","seed":202610314071,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":8,\"b\":5,\"d\":2,\"n\":12}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=8, b=5. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(0, 1, 4, 0, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 4, 0, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0072","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0a8ca54660f8edef898b702ff004481db13b26b42325293c99350d9160fc713","family_id":"qf-mod31-quotring-multiply","seed":202610314072,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":3,\"b\":7,\"d\":3,\"n\":18}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=3, b=7. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(0, 1, 3, 0, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 3, 0, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0073","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e45cf0506eb7a53fe6b72d12c3e424d49e31f391f7cb4b73dcdcdb9e030b2f8a","family_id":"qf-mod31-quotring-multiply","seed":202610314073,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":22,\"b\":21,\"d\":4,\"n\":24}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=22, b=21. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(2, 1, 6, 2, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (2, 1, 6, 2, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0074","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86653286929c4c6747146c2bb8db34dea1b2d40531525008f7665c064bca0c56","family_id":"qf-mod31-quotring-multiply","seed":202610314074,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":23,\"b\":17,\"d\":5,\"n\":30}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=23, b=17. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(3, 2, 1, 1, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 2, 1, 1, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0075","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"969ecc9332e47dcb60a94a534130499f141f345b1c5e9df8e08ed7e6e023281c","family_id":"qf-mod31-quotring-multiply","seed":202610314075,"semantic_template":"qf-mod31-quotring-multiply/{\"a\":24,\"b\":13,\"d\":6,\"n\":36}"},"fields":{"title":"Quotient Multiplication Must Respect the Same Ideal","stem":"Use the parent ring Z_36 with residues 0 through 35 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=24, b=13. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).","answer":{"kind":"expression","value":"(0, 1, 24, 0, [a][b]=[ab])","explanation":"Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 24, 0, [a][b]=[ab])."},"skills":["Quotient Multiplication Must Respect the Same Ideal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0076","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0b5d664a82c3730b19774f04ffcee4d360798bddc43ad278e56f8837f26225c","family_id":"qf-mod31-quotring-well-defined","seed":202610314076,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":0,\"a_prime\":2,\"b\":1,\"b_prime\":3,\"d\":2,\"n\":4}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_4 with residues 0 through 3 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=2, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([0, 1, 2, 3], [1, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3], [1, 1], [0, 0], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0077","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"282dc2a4d0b56a2938da449a0ac9795b0f391447dadd7040c6136ba37d171d0c","family_id":"qf-mod31-quotring-well-defined","seed":202610314077,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":1,\"a_prime\":4,\"b\":0,\"b_prime\":3,\"d\":3,\"n\":6}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=1, b=0, a_prime=4, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([1, 0, 4, 3], [1, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([1, 0, 4, 3], [1, 1], [0, 0], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0078","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df7f063d40150307eab74cac032d68c1716803ad81ce1d45a210678fdca50b9a","family_id":"qf-mod31-quotring-well-defined","seed":202610314078,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":2,\"a_prime\":6,\"b\":1,\"b_prime\":5,\"d\":4,\"n\":8}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=2, b=1, a_prime=6, b_prime=5 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([2, 1, 6, 5], [3, 3], [2, 2], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([2, 1, 6, 5], [3, 3], [2, 2], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0079","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68f80ae29c68ae01f1789320998978cc1b8b51f03b1fd9d32d17d6d9bdba3880","family_id":"qf-mod31-quotring-well-defined","seed":202610314079,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":3,\"a_prime\":8,\"b\":2,\"b_prime\":7,\"d\":5,\"n\":10}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=2, a_prime=8, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0080","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f14eac477bb5f81d8eb42567464fef0cbd604ecd2692d39d135ecd0dd48a803e","family_id":"qf-mod31-quotring-well-defined","seed":202610314080,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":4,\"a_prime\":10,\"b\":3,\"b_prime\":9,\"d\":6,\"n\":12}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=4, b=3, a_prime=10, b_prime=9 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([4, 3, 10, 9], [1, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([4, 3, 10, 9], [1, 1], [0, 0], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0081","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"166860d04a66078d0e8a3bff2521922597c55a45c4b367ad6d34cde931df3bdc","family_id":"qf-mod31-quotring-well-defined","seed":202610314081,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":1,\"a_prime\":3,\"b\":1,\"b_prime\":3,\"d\":2,\"n\":6}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=1, b=1, a_prime=3, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([1, 1, 3, 3], [0, 0], [1, 1], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([1, 1, 3, 3], [0, 0], [1, 1], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0082","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2e0175baa3a5ca48673a661b7dd38689f1b50e1d51e9796ba348da50cbfffad","family_id":"qf-mod31-quotring-well-defined","seed":202610314082,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":0,\"a_prime\":3,\"b\":1,\"b_prime\":4,\"d\":3,\"n\":9}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_9 with residues 0 through 8 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=3, b_prime=4 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([0, 1, 3, 4], [1, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 3, 4], [1, 1], [0, 0], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0083","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8241c40a4377a75690ed1cf5d20030a4b77ae90250fad09f68082a20d9a4fab","family_id":"qf-mod31-quotring-well-defined","seed":202610314083,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":3,\"a_prime\":7,\"b\":3,\"b_prime\":7,\"d\":4,\"n\":12}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=3, a_prime=7, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([3, 3, 7, 7], [2, 2], [1, 1], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 3, 7, 7], [2, 2], [1, 1], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0084","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"905715433cde6f88f40d54090e19dd0811840a00313edea9a9f6fca62b19f544","family_id":"qf-mod31-quotring-well-defined","seed":202610314084,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":3,\"a_prime\":8,\"b\":2,\"b_prime\":7,\"d\":5,\"n\":15}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=2, a_prime=8, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0085","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd6468e223c93c7549f4fd497e137a43c0b66998f0f288c8c7c75853f9bdcd7e","family_id":"qf-mod31-quotring-well-defined","seed":202610314085,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":3,\"a_prime\":9,\"b\":1,\"b_prime\":7,\"d\":6,\"n\":18}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=1, a_prime=9, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([3, 1, 9, 7], [4, 4], [3, 3], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 1, 9, 7], [4, 4], [3, 3], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0086","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f3579ad18b8a03c29791ddf22d24830a215d48ed050f564627f36818cba70b2","family_id":"qf-mod31-quotring-well-defined","seed":202610314086,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":0,\"a_prime\":2,\"b\":1,\"b_prime\":3,\"d\":2,\"n\":8}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=2, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([0, 1, 2, 3], [1, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3], [1, 1], [0, 0], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0087","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5f1f744b7ac33d59eb7524d8feab283869a160953687e945ccf8edd5f4dcfa0","family_id":"qf-mod31-quotring-well-defined","seed":202610314087,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":2,\"a_prime\":5,\"b\":2,\"b_prime\":5,\"d\":3,\"n\":12}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=2, b=2, a_prime=5, b_prime=5 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([2, 2, 5, 5], [1, 1], [1, 1], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([2, 2, 5, 5], [1, 1], [1, 1], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0088","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4a19d22fcd3933de30db7674910c32b202740cda873e3d3262704209fb2639a","family_id":"qf-mod31-quotring-well-defined","seed":202610314088,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":0,\"a_prime\":4,\"b\":1,\"b_prime\":5,\"d\":4,\"n\":16}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_16 with residues 0 through 15 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=4, b_prime=5 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([0, 1, 4, 5], [1, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 4, 5], [1, 1], [0, 0], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0089","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a7a7beccb0c1892e2341268ee91b2718f73a270062d7d909d3c10492e1fea23","family_id":"qf-mod31-quotring-well-defined","seed":202610314089,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":3,\"a_prime\":8,\"b\":2,\"b_prime\":7,\"d\":5,\"n\":20}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=2, a_prime=8, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0090","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed64507e36d87b764f3fc3fe591da25a9ab796acc6905b70b269e5ddc7bacb50","family_id":"qf-mod31-quotring-well-defined","seed":202610314090,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":2,\"a_prime\":8,\"b\":5,\"b_prime\":11,\"d\":6,\"n\":24}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=2, b=5, a_prime=8, b_prime=11 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([2, 5, 8, 11], [1, 1], [4, 4], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([2, 5, 8, 11], [1, 1], [4, 4], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0091","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7db42301e8bb94f85f42c5d212e8cbbcf908e5c9075b32178e1676faf3a782c","family_id":"qf-mod31-quotring-well-defined","seed":202610314091,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":1,\"a_prime\":3,\"b\":1,\"b_prime\":3,\"d\":2,\"n\":10}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=1, b=1, a_prime=3, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([1, 1, 3, 3], [0, 0], [1, 1], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([1, 1, 3, 3], [0, 0], [1, 1], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0092","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46e94b997ab1eeaa0d98cad470a533520999d54a89f9d1b705a9887d79a024fa","family_id":"qf-mod31-quotring-well-defined","seed":202610314092,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":1,\"a_prime\":4,\"b\":0,\"b_prime\":3,\"d\":3,\"n\":15}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=1, b=0, a_prime=4, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([1, 0, 4, 3], [1, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([1, 0, 4, 3], [1, 1], [0, 0], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0093","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"165119cdd12c87af7d0300e31200e827c616524c81ae4152371a9140dd9a3954","family_id":"qf-mod31-quotring-well-defined","seed":202610314093,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":1,\"a_prime\":5,\"b\":3,\"b_prime\":7,\"d\":4,\"n\":20}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=1, b=3, a_prime=5, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([1, 3, 5, 7], [0, 0], [3, 3], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([1, 3, 5, 7], [0, 0], [3, 3], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0094","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94762f8d564ab58422856f4539fdaeea666fd1e824e1a15e39c928e9d4de3f54","family_id":"qf-mod31-quotring-well-defined","seed":202610314094,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":3,\"a_prime\":8,\"b\":2,\"b_prime\":7,\"d\":5,\"n\":25}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_25 with residues 0 through 24 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=2, a_prime=8, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0095","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90188e82a6c2cd932303be52755545f119448ee681118fadadbe32e76a4b3d4b","family_id":"qf-mod31-quotring-well-defined","seed":202610314095,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":1,\"a_prime\":7,\"b\":3,\"b_prime\":9,\"d\":6,\"n\":30}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=1, b=3, a_prime=7, b_prime=9 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([1, 3, 7, 9], [4, 4], [3, 3], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([1, 3, 7, 9], [4, 4], [3, 3], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0096","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e469ced1af6d5c8f0646e643efde3f9617794666a99cf29103a4d6b2f038d89","family_id":"qf-mod31-quotring-well-defined","seed":202610314096,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":0,\"a_prime\":2,\"b\":1,\"b_prime\":3,\"d\":2,\"n\":12}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=2, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([0, 1, 2, 3], [1, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3], [1, 1], [0, 0], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0097","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"190c03c37a89521f93edfed19f96bfb5df118e7b8ad9026655128892716918de","family_id":"qf-mod31-quotring-well-defined","seed":202610314097,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":0,\"a_prime\":3,\"b\":1,\"b_prime\":4,\"d\":3,\"n\":18}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=3, b_prime=4 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([0, 1, 3, 4], [1, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 3, 4], [1, 1], [0, 0], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0098","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89764f13d48ff3eb3abc95a701a94087e30cd6e5f31872d69acbe8b04045ba94","family_id":"qf-mod31-quotring-well-defined","seed":202610314098,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":2,\"a_prime\":6,\"b\":1,\"b_prime\":5,\"d\":4,\"n\":24}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=2, b=1, a_prime=6, b_prime=5 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([2, 1, 6, 5], [3, 3], [2, 2], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([2, 1, 6, 5], [3, 3], [2, 2], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0099","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35ae698adcb9e07a882ef26d38e725c4912c6420fb9f66a4aa9e49c6ae6435c5","family_id":"qf-mod31-quotring-well-defined","seed":202610314099,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":3,\"a_prime\":8,\"b\":2,\"b_prime\":7,\"d\":5,\"n\":30}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=2, a_prime=8, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0100","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5b6ef90652d18ee25768a6ecb3503ecf65b32c83440b41465ca88027d826703","family_id":"qf-mod31-quotring-well-defined","seed":202610314100,"semantic_template":"qf-mod31-quotring-well-defined/{\"a\":0,\"a_prime\":6,\"b\":1,\"b_prime\":7,\"d\":6,\"n\":36}"},"fields":{"title":"Changing Both Representatives Must Preserve Both Operations","stem":"Use the parent ring Z_36 with residues 0 through 35 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=6, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).","answer":{"kind":"expression","value":"([0, 1, 6, 7], [1, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 6, 7], [1, 1], [0, 0], Yes, Yes, Yes)."},"skills":["Changing Both Representatives Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}