{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r02-arithmetic-operations-0001","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2dccc73796695b1a7972f6a7d14c739ec95465523a00da13a4eb8e8f72a93c9","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112001,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"1/2\",\"0/1\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=1/2, b=0/1. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 1/2, 0/1, undefined)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (1/2, 1/2, 0/1, undefined)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0002","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a36e46333a4a5e669477cff943b941725bb406b1de6e825b38eb9a5c4b43a34c","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112002,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"0/1\",\"-2/3\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=0/1, b=-2/3. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-2/3, 2/3, 0/1, 0/1)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-2/3, 2/3, 0/1, 0/1)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0003","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8eb40d5cff6350406cee1e0e9062ee373898b482b9c54e2664197beadfb88484","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112003,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-10/3\",\"3/4\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-10/3, b=3/4. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-31/12, -49/12, -5/2, -40/9)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-31/12, -49/12, -5/2, -40/9)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0004","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6b3b6cb438844d532aeb5e02be9dd16ca99c5d16db8b38a8dca49781a3770e1","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112004,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-9/1\",\"-4/5\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-9/1, b=-4/5. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-49/5, -41/5, 36/5, 45/4)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-49/5, -41/5, 36/5, 45/4)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0005","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"467325d163a22738c1546fa6c9bd0de76b278bbf4627f7f409fd3307e576d9f0","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112005,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-4/1\",\"5/2\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-8/2, b=5/2. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-3/2, -13/2, -10/1, -8/5)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-3/2, -13/2, -10/1, -8/5)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0006","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85f2ec31fa60dce8546a13abd140590cc9dfb9a55b912a4b45778f20ab3c493a","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112006,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-7/3\",\"-2/1\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-7/3, b=-6/3. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-13/3, -1/3, 14/3, 7/6)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-13/3, -1/3, 14/3, 7/6)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0007","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38fec3f16613c415fcb0c8e6aaf758a06de8937f721483af4c488b37598359b3","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112007,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-6/1\",\"7/4\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-6/1, b=7/4. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-17/4, -31/4, -21/2, -24/7)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-17/4, -31/4, -21/2, -24/7)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0008","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89a334f6ba7ab437665a57c0f6360e7c160ba3fc6e99f0e14c72e70bba7df45a","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112008,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-5/2\",\"-8/5\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-5/2, b=-8/5. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-41/10, -9/10, 4/1, 25/16)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-41/10, -9/10, 4/1, 25/16)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0009","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc21404e9d6f6a3da1c1607eeb41d2a4c1afbacd16b1d3a5a26553e650e87050","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112009,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-4/3\",\"9/2\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-4/3, b=9/2. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(19/6, -35/6, -6/1, -8/27)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (19/6, -35/6, -6/1, -8/27)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0010","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e154886b1dd2ca3a5ddfd8ad7ae5c7bd1422446fd2642d9b583d5627b607aa4b","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112010,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-3/1\",\"-10/3\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-3/1, b=-10/3. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-19/3, 1/3, 10/1, 9/10)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-19/3, 1/3, 10/1, 9/10)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0011","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00e2e8eb21033899230de59748b17994d4a5b80fd3b05387c309ffb735a58922","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112011,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-1/1\",\"11/4\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-2/2, b=11/4. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(7/4, -15/4, -11/4, -4/11)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (7/4, -15/4, -11/4, -4/11)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0012","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f411385b1505b59a309646d721c12bb5a5ee43a25c54462c3d1c07f7ccdfa09e","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112012,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-1/3\",\"-12/5\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-1/3, b=-12/5. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-41/15, 31/15, 4/5, 5/36)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-41/15, 31/15, 4/5, 5/36)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0013","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97d27ea5c5282373dd683e9279f500979bd07bafd2c5a9f506d3db4061c3ac82","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112013,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"0/1\",\"13/2\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=0/1, b=13/2. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(13/2, -13/2, 0/1, 0/1)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (13/2, -13/2, 0/1, 0/1)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0014","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90ce4c906791b52126563d838ec0e88903e06672574ab8f23cfde180dcb88aaa","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112014,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"1/2\",\"-14/3\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=1/2, b=-14/3. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-25/6, 31/6, -7/3, -3/28)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-25/6, 31/6, -7/3, -3/28)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0015","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd9e421100e8951a2362bfb06db6f66574c741731c3957223c088f1fb4f33501","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112015,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"2/3\",\"15/4\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=2/3, b=15/4. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(53/12, -37/12, 5/2, 8/45)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (53/12, -37/12, 5/2, 8/45)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0016","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac834e61c14436335f0a04b3ec150adf80f680baaddf8e862e5f9d91c049436b","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112016,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"3/1\",\"-16/5\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=3/1, b=-16/5. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/5, 31/5, -48/5, -15/16)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-1/5, 31/5, -48/5, -15/16)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0017","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c6b7b5a5c120a38eca75e936b8a4860e52207e24d5aebec21ee292b80c486be","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112017,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"2/1\",\"17/2\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=4/2, b=17/2. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(21/2, -13/2, 17/1, 4/17)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (21/2, -13/2, 17/1, 4/17)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0018","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3afd0889294d0527504649e4e590a70936c64a9ab6166447c8ab32e90248dcb","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112018,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"5/3\",\"-6/1\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=5/3, b=-18/3. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-13/3, 23/3, -10/1, -5/18)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-13/3, 23/3, -10/1, -5/18)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0019","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2bccb302222a099cc3a21d89a1da8ba5e5c2720f45c9d14af77b83a6f70d0ed","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112019,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"6/1\",\"19/4\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=6/1, b=19/4. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(43/4, 5/4, 57/2, 24/19)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (43/4, 5/4, 57/2, 24/19)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0020","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7554377cc53fec595fc6ae8909664c791af90f2203c1c1731e55db52a29c196","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112020,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"7/2\",\"-4/1\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=7/2, b=-20/5. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/2, 15/2, -14/1, -7/8)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-1/2, 15/2, -14/1, -7/8)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0021","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22071df6d5b0f84e98f66599ef7895a84cb93060755204f148da991040ccceed","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112021,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"8/3\",\"21/2\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=8/3, b=21/2. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(79/6, -47/6, 28/1, 16/63)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (79/6, -47/6, 28/1, 16/63)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0022","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7d7ee41879dcc21dc2e62154dd0d11bab2616712c051311120ffbefe62a940d","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112022,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"9/1\",\"-22/3\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=9/1, b=-22/3. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(5/3, 49/3, -66/1, -27/22)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (5/3, 49/3, -66/1, -27/22)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0023","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"911d468e37639bca3fb8f249525e2e343105347552d5316e67b03787098e9c58","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112023,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"5/1\",\"23/4\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=10/2, b=23/4. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(43/4, -3/4, 115/4, 20/23)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (43/4, -3/4, 115/4, 20/23)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0024","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6990869510b387ea6e164a8ee9ce25f2709d23af64a9b6630a99b4ab8b2d9cd6","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112024,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"11/3\",\"-24/5\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=11/3, b=-24/5. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-17/15, 127/15, -88/5, -55/72)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-17/15, 127/15, -88/5, -55/72)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0025","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ec4f01e73ce2da4be92cb4ef76682e1dd2eb6e324581bda0a5859f336218566","family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112025,"semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"12/1\",\"25/2\"]"},"fields":{"title":"Ordered Four Rational Operations","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=12/1, b=25/2. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(49/2, -1/2, 150/1, 24/25)","explanation":"Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (49/2, -1/2, 150/1, 24/25)."},"skills":["Ordered Four Rational 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-20261008-advance-20261011-r02-arithmetic-operations-0026","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23619d4e00452be1a2b0acf1c2a768456a1aa5523d21c55660edfdd4e06e87bd","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112026,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"zero_pair\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=0/1, b=2/1, c=0/1. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(0/1, 0/1)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (0/1, 0/1)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0027","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26254bce23cbd06a53a7c6436eee6e7d74689b773f9a6b23a4fde8378ae1ecd5","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112027,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"annihilated_outer\",\"3/2\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=3/2, b=9/5, c=0/1. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(0/1, 3/2)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (0/1, 3/2)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0028","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66e6087defd52e74eb893053690d7c36206a7b5e4a03fe0cdbf1853d81e4e8e3","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112028,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"neutral_multiplier\",\"1/1\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=1/3, b=2/3, c=1/1. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/1, 1/1)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (1/1, 1/1)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0029","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f78ab8ba5d77e7b41abc8cb47028b750a8d3a7265f31f95c60749b602d1537b5","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112029,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"equal_pair\",\"1/2\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=0/1, b=2/3, c=3/4. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 1/2)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (1/2, 1/2)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0030","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"819ba5d0fa40e2d8de8e38680cae27c83d4533974dcb092e6d05cdc597bbafd1","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112030,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"-4/1\",\"-1/1\",\"5/2\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-8/2, b=-3/3, c=5/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-25/2, -13/2)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-25/2, -13/2)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0031","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59e64e0e35bdb58f9d30a810ac00d89ae17f406505427db87808ebb2097b9a2e","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112031,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"-7/2\",\"-4/3\",\"3/1\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-7/2, b=-4/3, c=6/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-29/2, -15/2)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-29/2, -15/2)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0032","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5b4e18f98e72e56c9391792108b9cdb3e26b722f5b1bb8ee75b15e0e0e82115","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112032,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"-3/1\",\"-5/3\",\"7/2\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-6/2, b=-5/3, c=7/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-49/3, -53/6)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-49/3, -53/6)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0033","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f42cc85f8fca37d206f83364db33ec269ebe9fa02ba389050cbeab8ee1bea37c","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112033,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"-5/2\",\"-2/1\",\"4/1\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-5/2, b=-6/3, c=8/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-18/1, -21/2)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-18/1, -21/2)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0034","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a6298e6e4c345a236f1ea5b5977a7bc4c8fc9ce83770b632f327c9442deac48","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112034,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"-2/1\",\"-7/3\",\"9/2\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-4/2, b=-7/3, c=9/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-39/2, -25/2)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-39/2, -25/2)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0035","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f74db0f7dfe4e3c3d3c95c4ab76548a2c5d7885a70150d84d59d66fd67181b3f","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112035,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"-3/2\",\"-8/3\",\"5/1\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-3/2, b=-8/3, c=10/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-125/6, -89/6)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-125/6, -89/6)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0036","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"878f75bab0b23064a28fedfb46163c6664455a850193e6594a1aafcb688228bd","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112036,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"-1/1\",\"-3/1\",\"11/2\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-2/2, b=-9/3, c=11/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-22/1, -35/2)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-22/1, -35/2)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0037","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32f08e4567ab7ebce123daee3f8d1e3ae1c55f80d54dc5a4904ef1a2463ab4b7","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112037,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"-1/2\",\"-10/3\",\"6/1\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-1/2, b=-10/3, c=12/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-23/1, -41/2)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-23/1, -41/2)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0038","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"457120b0f227e4c02d2b38b928a2bbb75353fd2167d80b200a85ce5ca22018f2","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112038,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"equal_pair\",\"-143/6\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=0/2, b=-11/3, c=13/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-143/6, -143/6)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-143/6, -143/6)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0039","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2eb343c3fa7f001324ab3c07182c0faccfa01a64529e3b9dad77a6d154cc4140","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112039,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"1/2\",\"-4/1\",\"7/1\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=1/2, b=-12/3, c=14/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-49/2, -55/2)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-49/2, -55/2)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0040","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34df56192599a629aee4bb3f66681f2b697f78a664bd56934688bd949c1c5708","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112040,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"1/1\",\"-13/3\",\"15/2\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=2/2, b=-13/3, c=15/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-25/1, -63/2)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-25/1, -63/2)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0041","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"659caeada606b515beee67d4539a40164bf1553d0ab77679edee6c8bd324af49","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112041,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"3/2\",\"-14/3\",\"8/1\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=3/2, b=-14/3, c=16/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-76/3, -215/6)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-76/3, -215/6)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0042","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49bf9f4e48b5240e0eb36fdd56e798f5c4e16e2d833bb2f522e4f684c75508f4","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112042,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"2/1\",\"-5/1\",\"17/2\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=4/2, b=-15/3, c=17/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-51/2, -81/2)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-51/2, -81/2)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0043","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6af8a792dcdba38854dc6b875de4e4e923329c08b480992888c9aa77b5840eb8","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112043,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"5/2\",\"-16/3\",\"9/1\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=5/2, b=-16/3, c=18/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-51/2, -91/2)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-51/2, -91/2)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0044","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e96266c7bf85f885d2289a1648dfc55779187eb0489294f1c39593a275f4980c","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112044,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"3/1\",\"-17/3\",\"19/2\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=6/2, b=-17/3, c=19/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-76/3, -305/6)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-76/3, -305/6)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0045","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e15c5a60ce808d9260bfe9533b459430b2f9aae4fbd6ddaae2a6692f0904e918","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112045,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"7/2\",\"-6/1\",\"10/1\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=7/2, b=-18/3, c=20/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-25/1, -113/2)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-25/1, -113/2)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0046","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40880177b7306ab3b515a95698c05d547f03c5eef8ab6e9c9b1c158f79507386","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112046,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"4/1\",\"-19/3\",\"21/2\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=8/2, b=-19/3, c=21/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-49/2, -125/2)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-49/2, -125/2)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0047","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49e109f53c17e34a6fd79ac9daa9c85ced70b2ce226347d1798ce5f03863dbb4","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112047,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"9/2\",\"-20/3\",\"11/1\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=9/2, b=-20/3, c=22/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-143/6, -413/6)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-143/6, -413/6)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0048","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29366c941cfeba991e19479cbd2a5e4b53a64a12ed45f959259bb46169c9dd93","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112048,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"5/1\",\"-7/1\",\"23/2\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=10/2, b=-21/3, c=23/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-23/1, -151/2)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-23/1, -151/2)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0049","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c313baaeb1e6aa0a007c802e8dee7c8ff2194d30935d578dff136e0ac6e59089","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112049,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"11/2\",\"-22/3\",\"12/1\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=11/2, b=-22/3, c=24/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-22/1, -165/2)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-22/1, -165/2)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0050","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f07d79fbb4862ee682cf191e07ef80bd6dbc411c0d45f19a023aad0d372c076d","family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112050,"semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"6/1\",\"-23/3\",\"25/2\"]"},"fields":{"title":"Two Explicit Parenthesis Orders","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=12/2, b=-23/3, c=25/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-125/6, -539/6)","explanation":"The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-125/6, -539/6)."},"skills":["Two Explicit Parenthesis Orders"],"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-20261008-advance-20261011-r02-arithmetic-operations-0051","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d4dcc77cf0e5db4aef14a0e71179e494e3ba135d4bfc0dd8b2beaa04927817d","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112051,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"zero_operand\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=0/1, candidate r=7/2. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(No, -1/1)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -1/1)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0052","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39ec64706c2ad8a1fee0823b08bbf0c074946a43146a5a551a9425003e65368d","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112052,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"zero_candidate\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=2/3, candidate r=0/1. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(No, -1/1)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -1/1)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0053","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98253ba0535a90c2a7d7edfe9cc677f4dd889c7f4b17d3849a25104734b02f5e","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112053,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"1/4\",\"3/2\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=3/2, candidate r=1/4. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(No, -5/8)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -5/8)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0054","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa436068b868d60357990ee8da70060ef029a79dd7d2744577371e7bf4dce96b","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112054,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-1/2\",\"-2/1\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-4/2, candidate r=-2/4. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 0/1)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (Yes, 0/1)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0055","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb7d23b502ef6b8368e8677d432377a0b9c7628b11ff9b2a3bb7ff32c56748a2","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112055,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"1/6\",\"5/2\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=5/2, candidate r=1/6. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(No, -7/12)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -7/12)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0056","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec978123de58ea9381538f242819ad980510c4625f62efd981832cd032b9f640","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112056,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-3/1\",\"1/7\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-6/2, candidate r=1/7. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(No, -10/7)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -10/7)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0057","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab5ddd8fd0b9d96b8c9dbea28054322cf8073a196b2aa9478f2f973697563bdc","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112057,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"2/7\",\"7/2\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=7/2, candidate r=2/7. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 0/1)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (Yes, 0/1)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0058","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e0f1256b6e4d198abcfe6cb3d58e051f7636e3644c6800568c7541f6d5f95a5","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112058,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-4/1\",\"1/9\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-8/2, candidate r=1/9. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(No, -13/9)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -13/9)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0059","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3732161acf4fa14ff15cbdd43fe6f6f7e75e88a64472685fea9969e22b03244d","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112059,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"1/10\",\"9/2\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=9/2, candidate r=1/10. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(No, -11/20)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -11/20)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0060","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f94a6ae76d95ead1cdb01f59898e35bf7ba81fdf13d9a0c3683c105e152c6587","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112060,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-1/5\",\"-5/1\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-10/2, candidate r=-2/10. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 0/1)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (Yes, 0/1)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0061","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0c61a0ea27effdda510b87e77d869bfb3c042c187835a3c6617a2b03a04b733","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112061,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"1/12\",\"11/2\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=11/2, candidate r=1/12. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(No, -13/24)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -13/24)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0062","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79b7b7fc69b92a8ff84fe3efc49552a9ada64cefa06b2889da7716a857bf28fc","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112062,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-6/1\",\"1/13\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-12/2, candidate r=1/13. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(No, -19/13)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -19/13)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0063","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"113689563f21bec558b1706b3422c79d8943ab68f8de1fbce91960468ae42b5f","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112063,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"13/2\",\"2/13\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=13/2, candidate r=2/13. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 0/1)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (Yes, 0/1)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0064","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21acf5e4e05bcccf900e033838dba114c6b0020a8b0b8b2b339050638aed3c51","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112064,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-7/1\",\"1/15\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-14/2, candidate r=1/15. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(No, -22/15)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -22/15)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0065","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eedade513a195e82beb7f8d928ee8de6478fc9390652cf3528cf0e30c4f31e0e","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112065,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"1/16\",\"15/2\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=15/2, candidate r=1/16. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(No, -17/32)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -17/32)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0066","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bc1f1cb6244cad0dfef636757b5b8213baedec8e84d853d86af9fa8b11897e0","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112066,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-1/8\",\"-8/1\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-16/2, candidate r=-2/16. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 0/1)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (Yes, 0/1)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0067","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce9a397afbfa98823613a88f0d5e0c42c17abf85dc89ac40ae1f16bf0195542a","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112067,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"1/18\",\"17/2\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=17/2, candidate r=1/18. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(No, -19/36)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -19/36)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0068","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35f648039597ca289250b187520e91e80e40ec53533d4ce4bcedde6a115f5f86","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112068,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-9/1\",\"1/19\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-18/2, candidate r=1/19. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(No, -28/19)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -28/19)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0069","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4320b53cdeef61c032efa9580784c0922ca1ec7d483203b45dc0d509abbef4fe","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112069,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"19/2\",\"2/19\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=19/2, candidate r=2/19. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 0/1)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (Yes, 0/1)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0070","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4a73b6e63a1997f80cd4d2b732526aa5926c6c02b1b7460053882bcb96d6ad3","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112070,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-10/1\",\"1/21\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-20/2, candidate r=1/21. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(No, -31/21)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -31/21)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0071","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77c15cdc8bbdc3e84e22895b96b6139c1ed2ffc2cc1dbbc66787165f1689250c","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112071,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"1/22\",\"21/2\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=21/2, candidate r=1/22. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(No, -23/44)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -23/44)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0072","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2385dc112767f4bd4be33bec047b0a5f3e2fab5dc87d60be89ec4c61f0136aa","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112072,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-1/11\",\"-11/1\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-22/2, candidate r=-2/22. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 0/1)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (Yes, 0/1)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0073","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edfcd690e9b6411ffc704c87d7c7b32bfc9d29906268e3c2bae8fff2d844cdec","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112073,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"1/24\",\"23/2\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=23/2, candidate r=1/24. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(No, -25/48)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -25/48)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0074","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33d00e4fd1f86ce4e8ac0bd774a87cad049513cba56dec3524bf491e594c7a8b","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112074,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-12/1\",\"1/25\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-24/2, candidate r=1/25. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(No, -37/25)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -37/25)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0075","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0db4cc8f64b707ca65899ad2e655cc4fd63c428ec7f17ce7f968d6ad34b2bfff","family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112075,"semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"2/25\",\"25/2\"]"},"fields":{"title":"Inverse Candidate and Multiplicative Defect","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=25/2, candidate r=2/25. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 0/1)","explanation":"Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (Yes, 0/1)."},"skills":["Inverse Candidate and Multiplicative Defect"],"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-20261008-advance-20261011-r02-arithmetic-operations-0076","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eee63bdef55ee3287b9d6783e92cf0bf5fdd6f382aa520df382d1bdc71f3d770","family_id":"qf-density11-arithmetic-system-membership","seed":202610112076,"semantic_template":"qf-density11-arithmetic-system-membership/[0,0]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=0, b=0. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(0, Yes, undefined, undefined)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (0, Yes, undefined, undefined)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0077","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68cc177373a07a818cc537647a3d3e3073341518f9357148f6c0e65f60fd538a","family_id":"qf-density11-arithmetic-system-membership","seed":202610112077,"semantic_template":"qf-density11-arithmetic-system-membership/[1,0]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=1, b=0. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(1, Yes, undefined, undefined)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (1, Yes, undefined, undefined)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0078","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"422cdaec4256f6ac75ca2fd8b8570c9d06196f96c0ad1a21ebbde7a092607936","family_id":"qf-density11-arithmetic-system-membership","seed":202610112078,"semantic_template":"qf-density11-arithmetic-system-membership/[1,2]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=1, b=2. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(-1, No, 1/2, No)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (-1, No, 1/2, No)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0079","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41b8426102e4e5eabcbd726841d2bd1d1fec7ca8fb51444f2c2fe15dedf0859d","family_id":"qf-density11-arithmetic-system-membership","seed":202610112079,"semantic_template":"qf-density11-arithmetic-system-membership/[6,3]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=6, b=3. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(3, Yes, 2/1, Yes)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (3, Yes, 2/1, Yes)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0080","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5143ec9fad3898da99228d5d15f50e9355b6473a980883bea5c078cee79cee20","family_id":"qf-density11-arithmetic-system-membership","seed":202610112080,"semantic_template":"qf-density11-arithmetic-system-membership/[0,3]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=0, b=3. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(-3, No, 0/1, Yes)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (-3, No, 0/1, Yes)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0081","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c6fd02e5f6d39d896e132dbafb7a22e8f13eb4d0a7fc29a34f23108e5ccd5d8","family_id":"qf-density11-arithmetic-system-membership","seed":202610112081,"semantic_template":"qf-density11-arithmetic-system-membership/[2,1]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=2, b=1. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(1, Yes, 2/1, Yes)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (1, Yes, 2/1, Yes)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0082","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d040d61201d866c2cd492a1e558826ee5709c6e797f0552d6cb1254d048d7bc4","family_id":"qf-density11-arithmetic-system-membership","seed":202610112082,"semantic_template":"qf-density11-arithmetic-system-membership/[3,2]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=3, b=2. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(1, Yes, 3/2, No)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (1, Yes, 3/2, No)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0083","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a384508493057ef31b7859438fa98799f858d219e901f8045b7b7fe4aabda273","family_id":"qf-density11-arithmetic-system-membership","seed":202610112083,"semantic_template":"qf-density11-arithmetic-system-membership/[4,3]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=4, b=3. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(1, Yes, 4/3, No)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (1, Yes, 4/3, No)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0084","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b62c97ad4d16461cd6733b9002c7e09e170e37f941b1dc25dbf55a3ea04bfdd4","family_id":"qf-density11-arithmetic-system-membership","seed":202610112084,"semantic_template":"qf-density11-arithmetic-system-membership/[5,4]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=5, b=4. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(1, Yes, 5/4, No)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (1, Yes, 5/4, No)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0085","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88518e34b1695f9cdf3313717cb096be42cf88e016a411cca51e148ebec8b2f7","family_id":"qf-density11-arithmetic-system-membership","seed":202610112085,"semantic_template":"qf-density11-arithmetic-system-membership/[6,5]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=6, b=5. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(1, Yes, 6/5, No)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (1, Yes, 6/5, No)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0086","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3153c9d446da9e79b1131438035cad43e150795b6bb5fff255e7ccd1c867ab7a","family_id":"qf-density11-arithmetic-system-membership","seed":202610112086,"semantic_template":"qf-density11-arithmetic-system-membership/[7,1]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=7, b=1. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(6, Yes, 7/1, Yes)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (6, Yes, 7/1, Yes)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0087","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77d8b8c223c0c4f16bc10f07fb105c3d6963a52eedbde346de6f2a3d3eaa69d2","family_id":"qf-density11-arithmetic-system-membership","seed":202610112087,"semantic_template":"qf-density11-arithmetic-system-membership/[8,2]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=8, b=2. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(6, Yes, 4/1, Yes)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (6, Yes, 4/1, Yes)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0088","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8ea4c61dbd254ecc395cc8685ae778fe25fc9a1589d8c030fda3c981b670d20","family_id":"qf-density11-arithmetic-system-membership","seed":202610112088,"semantic_template":"qf-density11-arithmetic-system-membership/[9,3]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=9, b=3. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(6, Yes, 3/1, Yes)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (6, Yes, 3/1, Yes)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0089","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23fa51b0222cc6326c09fcfe596870a4f52d7cf5e68534f978c7d60630760b27","family_id":"qf-density11-arithmetic-system-membership","seed":202610112089,"semantic_template":"qf-density11-arithmetic-system-membership/[10,4]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=10, b=4. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(6, Yes, 5/2, No)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (6, Yes, 5/2, No)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0090","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d39b4195afcdad1619dea090b3deedd04107f143a8815d083890c68ee86e94fe","family_id":"qf-density11-arithmetic-system-membership","seed":202610112090,"semantic_template":"qf-density11-arithmetic-system-membership/[11,5]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=11, b=5. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(6, Yes, 11/5, No)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (6, Yes, 11/5, No)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0091","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b0ed77e11d07bd4f6cdab9d8eaaa22dad57afbc86f29294497b0f00199ed66d","family_id":"qf-density11-arithmetic-system-membership","seed":202610112091,"semantic_template":"qf-density11-arithmetic-system-membership/[12,1]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=12, b=1. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(11, Yes, 12/1, Yes)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (11, Yes, 12/1, Yes)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0092","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fab2960f0f4ff02de5fd0d7c8e41714ac258d24460d7003c8b1d390d18bd76b5","family_id":"qf-density11-arithmetic-system-membership","seed":202610112092,"semantic_template":"qf-density11-arithmetic-system-membership/[13,2]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=13, b=2. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(11, Yes, 13/2, No)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (11, Yes, 13/2, No)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0093","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"785753865c032e34b06605204126861579c445cb81bb13ad25b79525c79c1df9","family_id":"qf-density11-arithmetic-system-membership","seed":202610112093,"semantic_template":"qf-density11-arithmetic-system-membership/[14,3]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=14, b=3. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(11, Yes, 14/3, No)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (11, Yes, 14/3, No)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0094","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45da3361bcf35a5f8ca5ead249ea8e65484fd5289cbf172dc6f08b9a6f74d627","family_id":"qf-density11-arithmetic-system-membership","seed":202610112094,"semantic_template":"qf-density11-arithmetic-system-membership/[15,4]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=15, b=4. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(11, Yes, 15/4, No)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (11, Yes, 15/4, No)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0095","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44d527369ffa52b37c7ab8aa91efeb1a868279f75cb2a40dda1341ef22a6713e","family_id":"qf-density11-arithmetic-system-membership","seed":202610112095,"semantic_template":"qf-density11-arithmetic-system-membership/[16,5]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=16, b=5. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(11, Yes, 16/5, No)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (11, Yes, 16/5, No)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0096","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6805cdee122139b47398a865a146edeae299d676105c71b4ae3a49167e15c444","family_id":"qf-density11-arithmetic-system-membership","seed":202610112096,"semantic_template":"qf-density11-arithmetic-system-membership/[17,1]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=17, b=1. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(16, Yes, 17/1, Yes)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (16, Yes, 17/1, Yes)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0097","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"100fb8abb2690e8a4d55c4737c2a2385686e23d452e64642b67b8e6674f57718","family_id":"qf-density11-arithmetic-system-membership","seed":202610112097,"semantic_template":"qf-density11-arithmetic-system-membership/[18,2]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=18, b=2. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(16, Yes, 9/1, Yes)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (16, Yes, 9/1, Yes)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0098","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05518157cc7da2112e5e3b0490c998129679905179ba5e18d914a4568e9c2b80","family_id":"qf-density11-arithmetic-system-membership","seed":202610112098,"semantic_template":"qf-density11-arithmetic-system-membership/[19,3]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=19, b=3. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(16, Yes, 19/3, No)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (16, Yes, 19/3, No)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0099","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8598098ad4b5a243587730593528a5d9dd20d46895aca3ab888a65da82375edb","family_id":"qf-density11-arithmetic-system-membership","seed":202610112099,"semantic_template":"qf-density11-arithmetic-system-membership/[20,4]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=20, b=4. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(16, Yes, 5/1, Yes)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (16, Yes, 5/1, Yes)."},"skills":["Q Results and Natural-System Eligibility"],"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-20261008-advance-20261011-r02-arithmetic-operations-0100","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83df5f9425a5027fdbf74be5298495299229e11c3be4bfd53c007b1f8a485c46","family_id":"qf-density11-arithmetic-system-membership","seed":202610112100,"semantic_template":"qf-density11-arithmetic-system-membership/[21,5]"},"fields":{"title":"Q Results and Natural-System Eligibility","stem":"All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=21, b=5. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.","answer":{"kind":"expression","value":"(16, Yes, 21/5, No)","explanation":"A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (16, Yes, 21/5, No)."},"skills":["Q Results and Natural-System Eligibility"],"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."}}