{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r02-integer-0001","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69daadd666e5d808b7367789c8057f25ea9aecb28362b0983d707f1af0a86421","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132001,"semantic_template":"qf-density13-integer-euclidean-divmod/[\"zero_dividend\"]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=0, positive b=5. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(0, 0)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (0, 0)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0002","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37163a4039eed4afca310a8235c9744cf6a64f5c3c068ba53f56cfb390e075a8","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132002,"semantic_template":"qf-density13-integer-euclidean-divmod/[-7,3]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=-7, positive b=3. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(-3, 2)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-3, 2)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0003","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0cf7a7341329762d632dd2ccf7c8a52d885ee0b7c83c4d6ba2f9455b6881a19","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132003,"semantic_template":"qf-density13-integer-euclidean-divmod/[3,4]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=3, positive b=4. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(0, 3)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (0, 3)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0004","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef1a11870ba1f85ad596767a63f55ee9b25c07a0154f0951f6b6123aa1fb6c26","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132004,"semantic_template":"qf-density13-integer-euclidean-divmod/[-4,5]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=-4, positive b=5. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(-1, 1)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-1, 1)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0005","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b5b858921302efe81c3cd419c89a6c7256008227f852d9f7edb27d1d51e375a","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132005,"semantic_template":"qf-density13-integer-euclidean-divmod/[5,6]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=5, positive b=6. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(0, 5)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (0, 5)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0006","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"239391d769ff2aa8ebf428766bef3dabe9f9127fdc8b02531f19ee067f6b87e5","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132006,"semantic_template":"qf-density13-integer-euclidean-divmod/[-6,2]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=-6, positive b=2. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(-3, 0)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-3, 0)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0007","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57002571c16946711e21ced6a64579efdeac2e58814f1531a44bcdc36624ef5e","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132007,"semantic_template":"qf-density13-integer-euclidean-divmod/[7,3]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=7, positive b=3. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(2, 1)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (2, 1)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0008","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ce216afd5806714503b98ec2ca425a615936130dfa6fca5f6944188254ca951","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132008,"semantic_template":"qf-density13-integer-euclidean-divmod/[-8,4]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=-8, positive b=4. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(-2, 0)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-2, 0)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0009","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66945fb5a5b9bc3eadcd840f948d3a60bc3056b1c3a71161cb94971ac1217ac5","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132009,"semantic_template":"qf-density13-integer-euclidean-divmod/[9,5]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=9, positive b=5. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(1, 4)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (1, 4)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0010","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c3374fd95a5cb8c70b0c4e19944a2d23a409d51c3609a434fe62fc77d21297e","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132010,"semantic_template":"qf-density13-integer-euclidean-divmod/[-10,6]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=-10, positive b=6. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(-2, 2)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-2, 2)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0011","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"332332699794002df9b42947a2e4f5d8f4734241e6d2edbc6717e1d31045b229","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132011,"semantic_template":"qf-density13-integer-euclidean-divmod/[11,2]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=11, positive b=2. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(5, 1)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (5, 1)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0012","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93bb79c4c3379f6b13842bd2e472919b7e796037b8b5665f3f93c8d018f68aa1","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132012,"semantic_template":"qf-density13-integer-euclidean-divmod/[-12,3]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=-12, positive b=3. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(-4, 0)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-4, 0)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0013","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a93473dfb499044281ccf327a61ebdfd673117e96d40821ba2aafbed03f2837","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132013,"semantic_template":"qf-density13-integer-euclidean-divmod/[13,4]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=13, positive b=4. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(3, 1)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (3, 1)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0014","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b19f29e238ffd818e4021574d0f3e3bac52b6c6ed8d7e27e4e23c94c18d26f1","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132014,"semantic_template":"qf-density13-integer-euclidean-divmod/[-14,5]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=-14, positive b=5. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(-3, 1)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-3, 1)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0015","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdf71893fd4175c8b86f3c76e7f7d3c37ae63c00e676070248c1ba44859a7e9c","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132015,"semantic_template":"qf-density13-integer-euclidean-divmod/[15,6]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=15, positive b=6. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(2, 3)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (2, 3)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0016","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd85c28178f7a47e168831daf0f9ae0fc74fcf7c76c95a50be3885f769ce5455","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132016,"semantic_template":"qf-density13-integer-euclidean-divmod/[-16,2]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=-16, positive b=2. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(-8, 0)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-8, 0)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0017","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfc303fa2ed963959138ff5493282ed3035e479059442572393ab8a1332acb4d","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132017,"semantic_template":"qf-density13-integer-euclidean-divmod/[17,3]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=17, positive b=3. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(5, 2)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (5, 2)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0018","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9658868078177813f7b1613d6208cb066aaaa7b44e0e337d53495ec22afbba32","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132018,"semantic_template":"qf-density13-integer-euclidean-divmod/[-18,4]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=-18, positive b=4. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(-5, 2)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-5, 2)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0019","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89e84ba8a752ecb14992a6cdcbf3483d6ac055f5a3d2e4d31d3c694270c933a3","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132019,"semantic_template":"qf-density13-integer-euclidean-divmod/[19,5]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=19, positive b=5. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(3, 4)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (3, 4)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0020","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f11b39423f06bf1ed61b52cd23c798351a2faadcd2940276a53441e9afc25342","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132020,"semantic_template":"qf-density13-integer-euclidean-divmod/[-20,6]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=-20, positive b=6. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(-4, 4)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-4, 4)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0021","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b40b2dd64541a66a9b1cc656665060af01af4ae8a67cb867e4a73f943e51580","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132021,"semantic_template":"qf-density13-integer-euclidean-divmod/[21,2]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=21, positive b=2. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(10, 1)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (10, 1)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0022","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05391a38e12d3636b8a51f7406283c8244d8743aed7b6bd015003e9898db0ecb","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132022,"semantic_template":"qf-density13-integer-euclidean-divmod/[-22,3]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=-22, positive b=3. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(-8, 2)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-8, 2)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0023","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c4df26c225478661756019b2edf641f05ba4776fe170f1e7bf10ccebe4505da","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132023,"semantic_template":"qf-density13-integer-euclidean-divmod/[23,4]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=23, positive b=4. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(5, 3)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (5, 3)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0024","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e86c7db749d1252294c17072a0fda39af7c02f9b98302602f1e9d08aab7ae03f","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132024,"semantic_template":"qf-density13-integer-euclidean-divmod/[-24,5]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=-24, positive b=5. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(-5, 1)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-5, 1)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0025","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0677a340c04d30b57a41c02e6483c4393f2cc787bff78ce5bea3bfcb5fdb430c","family_id":"qf-density13-integer-euclidean-divmod","seed":202610132025,"semantic_template":"qf-density13-integer-euclidean-divmod/[25,6]"},"fields":{"title":"Euclidean Integer Quotient and Remainder","stem":"Integer a=25, positive b=6. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.","answer":{"kind":"expression","value":"(4, 1)","explanation":"If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (4, 1)."},"skills":["Euclidean Integer Quotient and Remainder"],"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-20261009-advance-20261013-r02-integer-0026","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d8cfd8864e3f3936bfddb5f108983c7632b1e61088c2f5625136984b4236961","family_id":"qf-density13-integer-two-step-displacement","seed":202610132026,"semantic_template":"qf-density13-integer-two-step-displacement/[0,\"0\",\"0\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=0, signed steps s1=0, s2=0. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(0, 0, 0)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (0, 0, 0)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0027","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb9ec46ad84525725933e5c4ed69473e481886f1bf6555ff6f35297833259dde","family_id":"qf-density13-integer-two-step-displacement","seed":202610132027,"semantic_template":"qf-density13-integer-two-step-displacement/[1,\"0\",\"6\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=1, signed steps s1=3, s2=-3. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(1, 0, 6)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (1, 0, 6)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0028","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"547d00f393384f163d7a92656b587b6ba0a4f04731a2833065b12ee4dcc6df92","family_id":"qf-density13-integer-two-step-displacement","seed":202610132028,"semantic_template":"qf-density13-integer-two-step-displacement/[-10,\"0\",\"6\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=-10, signed steps s1=3, s2=-3. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(-10, 0, 6)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (-10, 0, 6)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0029","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ec33d219a9bea4acee424fba566d03ecf8351d793b82a9eaf1895f5e456ad64","family_id":"qf-density13-integer-two-step-displacement","seed":202610132029,"semantic_template":"qf-density13-integer-two-step-displacement/[-9,\"0\",\"8\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=-9, signed steps s1=4, s2=-4. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(-9, 0, 8)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (-9, 0, 8)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0030","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0199fcd06ad034af40248827e2459a4722c82fed9392a8d3ad58c613a79fa77","family_id":"qf-density13-integer-two-step-displacement","seed":202610132030,"semantic_template":"qf-density13-integer-two-step-displacement/[-8,\"0\",\"10\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=-8, signed steps s1=5, s2=-5. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(-8, 0, 10)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (-8, 0, 10)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0031","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"972a67ae0d6727641f92c90decabf44c0c3346d290bf146a5e2c9918cb68bfd4","family_id":"qf-density13-integer-two-step-displacement","seed":202610132031,"semantic_template":"qf-density13-integer-two-step-displacement/[-7,\"5\",\"7\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=-7, signed steps s1=6, s2=-1. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(-2, 5, 7)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (-2, 5, 7)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0032","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0a4283117eb8a2be6fcc3f3567bfc8a46d5d085dd36a60d5323a5a4b34cc8ba","family_id":"qf-density13-integer-two-step-displacement","seed":202610132032,"semantic_template":"qf-density13-integer-two-step-displacement/[-6,\"5\",\"9\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=-6, signed steps s1=7, s2=-2. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(-1, 5, 9)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (-1, 5, 9)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0033","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"522249749d3cad7f58d14d899ec6d1c3b7457265e7aa8d2fecd9812e52d0ddb5","family_id":"qf-density13-integer-two-step-displacement","seed":202610132033,"semantic_template":"qf-density13-integer-two-step-displacement/[-5,\"5\",\"11\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=-5, signed steps s1=8, s2=-3. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(0, 5, 11)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (0, 5, 11)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0034","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7aa20f4474398ca83eaf67f5d3688d13e5ca2c5d5dd136860895199b5f6cf28","family_id":"qf-density13-integer-two-step-displacement","seed":202610132034,"semantic_template":"qf-density13-integer-two-step-displacement/[-4,\"5\",\"13\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=-4, signed steps s1=9, s2=-4. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(1, 5, 13)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (1, 5, 13)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0035","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"283aaffb34d12cd767ad3f42f3751b12b8992209b1f2e8a54fb91c8a9f53a759","family_id":"qf-density13-integer-two-step-displacement","seed":202610132035,"semantic_template":"qf-density13-integer-two-step-displacement/[-3,\"5\",\"15\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=-3, signed steps s1=10, s2=-5. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(2, 5, 15)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (2, 5, 15)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0036","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4372ffdfab911bc74d237a245378e23d631807724eae73d6b8b191c5a09819db","family_id":"qf-density13-integer-two-step-displacement","seed":202610132036,"semantic_template":"qf-density13-integer-two-step-displacement/[-2,\"10\",\"12\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=-2, signed steps s1=11, s2=-1. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(8, 10, 12)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (8, 10, 12)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0037","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c9b82ac4b5706d20c45433ef007d887846e3ae3f15564653ae6ad4d3a778981","family_id":"qf-density13-integer-two-step-displacement","seed":202610132037,"semantic_template":"qf-density13-integer-two-step-displacement/[-1,\"10\",\"14\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=-1, signed steps s1=12, s2=-2. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(9, 10, 14)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (9, 10, 14)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0038","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f65d8cbc14709cd2a4b2f75addf3556b7ca8a62246f8069c996c1f27c58f03f4","family_id":"qf-density13-integer-two-step-displacement","seed":202610132038,"semantic_template":"qf-density13-integer-two-step-displacement/[0,\"10\",\"16\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=0, signed steps s1=13, s2=-3. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(10, 10, 16)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (10, 10, 16)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0039","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a00054f080c2331ff97466443e427f22ad48df06390bb86fe1c1388f9e0124a","family_id":"qf-density13-integer-two-step-displacement","seed":202610132039,"semantic_template":"qf-density13-integer-two-step-displacement/[1,\"10\",\"18\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=1, signed steps s1=14, s2=-4. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(11, 10, 18)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (11, 10, 18)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0040","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5dd7855eb293e62030449d98e853093717f59813644d3b7e8bb1d6cbdd23802e","family_id":"qf-density13-integer-two-step-displacement","seed":202610132040,"semantic_template":"qf-density13-integer-two-step-displacement/[2,\"10\",\"20\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=2, signed steps s1=15, s2=-5. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(12, 10, 20)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (12, 10, 20)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0041","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"223affb8880b48e3927eca21548e281cb9f5f66da4288c3ef07081b65d009127","family_id":"qf-density13-integer-two-step-displacement","seed":202610132041,"semantic_template":"qf-density13-integer-two-step-displacement/[3,\"15\",\"17\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=3, signed steps s1=16, s2=-1. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(18, 15, 17)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (18, 15, 17)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0042","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0248351c40c9039a47405427b8bf302391e3acd87a70cea69c9b135f0f00cb6d","family_id":"qf-density13-integer-two-step-displacement","seed":202610132042,"semantic_template":"qf-density13-integer-two-step-displacement/[4,\"15\",\"19\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=4, signed steps s1=17, s2=-2. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(19, 15, 19)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (19, 15, 19)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0043","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"747d4f1ed5275d39cfc606c42ab8b6cbd3d92fe756663fcaf7a6591b54716db1","family_id":"qf-density13-integer-two-step-displacement","seed":202610132043,"semantic_template":"qf-density13-integer-two-step-displacement/[5,\"15\",\"21\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=5, signed steps s1=18, s2=-3. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(20, 15, 21)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (20, 15, 21)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0044","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"701a237b6f5ddce4e5bc71b7fdac079e47925e05af985b7395cc1bd60b196b1b","family_id":"qf-density13-integer-two-step-displacement","seed":202610132044,"semantic_template":"qf-density13-integer-two-step-displacement/[6,\"15\",\"23\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=6, signed steps s1=19, s2=-4. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(21, 15, 23)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (21, 15, 23)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0045","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5792656a547ffa903527e5898635d0d4835939e1ab76aba907e0505fc92e3b2e","family_id":"qf-density13-integer-two-step-displacement","seed":202610132045,"semantic_template":"qf-density13-integer-two-step-displacement/[7,\"15\",\"25\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=7, signed steps s1=20, s2=-5. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(22, 15, 25)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (22, 15, 25)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0046","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"554699c2800ef65dc4ea6119c2b33493c1fbe19ee7b6b523bf9805edd65669ec","family_id":"qf-density13-integer-two-step-displacement","seed":202610132046,"semantic_template":"qf-density13-integer-two-step-displacement/[8,\"20\",\"22\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=8, signed steps s1=21, s2=-1. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(28, 20, 22)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (28, 20, 22)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0047","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21344a05f04e0e9b588550f31745b412a3bdea935e81327c5bda20cbd7d87e2c","family_id":"qf-density13-integer-two-step-displacement","seed":202610132047,"semantic_template":"qf-density13-integer-two-step-displacement/[9,\"20\",\"24\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=9, signed steps s1=22, s2=-2. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(29, 20, 24)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (29, 20, 24)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0048","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b12f982cf311c53a15ff86acf595b6cb701f8f3337d2e88786b2a066debf8ac5","family_id":"qf-density13-integer-two-step-displacement","seed":202610132048,"semantic_template":"qf-density13-integer-two-step-displacement/[10,\"20\",\"26\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=10, signed steps s1=23, s2=-3. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(30, 20, 26)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (30, 20, 26)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0049","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60f3b63d523d6a4508002d8ffb5b9e05de26c0a8103f546b1b3461867f326b5a","family_id":"qf-density13-integer-two-step-displacement","seed":202610132049,"semantic_template":"qf-density13-integer-two-step-displacement/[11,\"20\",\"28\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=11, signed steps s1=24, s2=-4. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(31, 20, 28)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (31, 20, 28)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0050","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46334affd59f9f13607e6259fd8f774b9ba1f2744ef295e51a925c01c2205f1c","family_id":"qf-density13-integer-two-step-displacement","seed":202610132050,"semantic_template":"qf-density13-integer-two-step-displacement/[12,\"20\",\"30\"]"},"fields":{"title":"Endpoint, Net Displacement, and Step Magnitude","stem":"Integer start=12, signed steps s1=25, s2=-5. Give (endpoint, s1+s2, |s1|+|s2|).","answer":{"kind":"expression","value":"(32, 20, 30)","explanation":"Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (32, 20, 30)."},"skills":["Endpoint, Net Displacement, and Step Magnitude"],"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-20261009-advance-20261013-r02-integer-0051","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ef503ee2c6ba9cf807616875870f2df73ed034240a97d511003c928b44608d0","family_id":"qf-density13-integer-opposite-difference","seed":202610132051,"semantic_template":"qf-density13-integer-opposite-difference/[0,0]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=0, b=0. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(0, 0, 0, 0)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (0, 0, 0, 0)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0052","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1fe00719c18225bedd655b75467cd8fd8b26d96c4d7f3bb80880d07cabb10ad","family_id":"qf-density13-integer-opposite-difference","seed":202610132052,"semantic_template":"qf-density13-integer-opposite-difference/[-11,0]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=-11, b=0. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(11, 0, -11, 11)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (11, 0, -11, 11)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0053","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"deda7ab48b7c20949532de76b5dd95e181c27cd6526c40aab4d101d2654bf8d4","family_id":"qf-density13-integer-opposite-difference","seed":202610132053,"semantic_template":"qf-density13-integer-opposite-difference/[-10,-1]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=-10, b=-1. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(10, 1, -9, 9)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (10, 1, -9, 9)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0054","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b305b793381a52e72d1b82dbb10c33215d65039d4395e1fdddc046b6ad6ecad","family_id":"qf-density13-integer-opposite-difference","seed":202610132054,"semantic_template":"qf-density13-integer-opposite-difference/[-9,-2]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=-9, b=-2. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(9, 2, -7, 7)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (9, 2, -7, 7)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0055","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8530be8caf7e429cb50e43339e7796534a46385783b7ac2ef48720ff99879b6","family_id":"qf-density13-integer-opposite-difference","seed":202610132055,"semantic_template":"qf-density13-integer-opposite-difference/[-8,-3]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=-8, b=-3. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(8, 3, -5, 5)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (8, 3, -5, 5)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0056","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7541fd323ec8d29408909e9bc197ed121d8d01ae13b564f0a6ff9d249884c5f","family_id":"qf-density13-integer-opposite-difference","seed":202610132056,"semantic_template":"qf-density13-integer-opposite-difference/[-7,-4]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=-7, b=-4. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(7, 4, -3, 3)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (7, 4, -3, 3)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0057","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e9120e3c399447611a20b72ada9653679ab94f086b8f719c83e297b89905952","family_id":"qf-density13-integer-opposite-difference","seed":202610132057,"semantic_template":"qf-density13-integer-opposite-difference/[-6,-5]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=-6, b=-5. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(6, 5, -1, 1)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (6, 5, -1, 1)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0058","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ceb7fd492cf4cdd279391f4badb126599fa6c25da698c96d46573355911da5b","family_id":"qf-density13-integer-opposite-difference","seed":202610132058,"semantic_template":"qf-density13-integer-opposite-difference/[-5,-6]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=-5, b=-6. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(5, 6, 1, -1)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (5, 6, 1, -1)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0059","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76cf1116d5871b03d4a732ef3b03798307aa5717dbc0c43ec02e9e6a67c96607","family_id":"qf-density13-integer-opposite-difference","seed":202610132059,"semantic_template":"qf-density13-integer-opposite-difference/[-4,-7]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=-4, b=-7. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(4, 7, 3, -3)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (4, 7, 3, -3)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0060","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3677373d4895cf3aba4a667804d636ac8e73b108bba83d0f725fa01c6b8c6e7","family_id":"qf-density13-integer-opposite-difference","seed":202610132060,"semantic_template":"qf-density13-integer-opposite-difference/[-3,-8]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=-3, b=-8. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(3, 8, 5, -5)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (3, 8, 5, -5)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0061","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3b26c129b982af049d19cd2530c19d6a30a1f817ba9f732bc0f48f3f0fcdc20","family_id":"qf-density13-integer-opposite-difference","seed":202610132061,"semantic_template":"qf-density13-integer-opposite-difference/[-2,-9]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=-2, b=-9. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(2, 9, 7, -7)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (2, 9, 7, -7)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0062","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53a14f93bdec71160f5ec814745a59103d0ef7fbbc456b918d03a74894a5c26f","family_id":"qf-density13-integer-opposite-difference","seed":202610132062,"semantic_template":"qf-density13-integer-opposite-difference/[-1,-10]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=-1, b=-10. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(1, 10, 9, -9)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (1, 10, 9, -9)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0063","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e00e18360417f0c553c2defe426f3e8c7da6fc297760bea33b4cd32f1b6d98ed","family_id":"qf-density13-integer-opposite-difference","seed":202610132063,"semantic_template":"qf-density13-integer-opposite-difference/[0,-11]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=0, b=-11. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(0, 11, 11, -11)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (0, 11, 11, -11)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0064","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1b876f0b743a3af6f0c1186ae73ae0ab04c28671476c4bf14cc93b28c9c73ac","family_id":"qf-density13-integer-opposite-difference","seed":202610132064,"semantic_template":"qf-density13-integer-opposite-difference/[1,-12]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=1, b=-12. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(-1, 12, 13, -13)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-1, 12, 13, -13)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0065","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e5dcadded693029b970f496ecc6ff5c97f0a178ad64f09b11624181f82166a9","family_id":"qf-density13-integer-opposite-difference","seed":202610132065,"semantic_template":"qf-density13-integer-opposite-difference/[2,-13]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=2, b=-13. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(-2, 13, 15, -15)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-2, 13, 15, -15)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0066","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a457637565fc641280f67cc5ab66e11af7fc914275c7b270af44b3ff60ee4ce","family_id":"qf-density13-integer-opposite-difference","seed":202610132066,"semantic_template":"qf-density13-integer-opposite-difference/[3,-14]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=3, b=-14. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(-3, 14, 17, -17)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-3, 14, 17, -17)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0067","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e6528c66111d7d8cbdfc9e85743bc3079abf4786f1f2f414f7cab97088acc2c","family_id":"qf-density13-integer-opposite-difference","seed":202610132067,"semantic_template":"qf-density13-integer-opposite-difference/[4,-15]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=4, b=-15. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(-4, 15, 19, -19)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-4, 15, 19, -19)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0068","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de100029b9e4dfc9e471ec7e75006cc6bfdfa60c285bb50845aab5f5c0faee61","family_id":"qf-density13-integer-opposite-difference","seed":202610132068,"semantic_template":"qf-density13-integer-opposite-difference/[5,-16]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=5, b=-16. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(-5, 16, 21, -21)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-5, 16, 21, -21)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0069","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b46120c837a25088628a34de941c34bdf033b83a655c90232eaabe3af002de7a","family_id":"qf-density13-integer-opposite-difference","seed":202610132069,"semantic_template":"qf-density13-integer-opposite-difference/[6,-17]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=6, b=-17. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(-6, 17, 23, -23)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-6, 17, 23, -23)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0070","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9ad66ce4e9ad3cfe93053a828b8411f44e1f26d782a7d59ae76edd0fb24ed34","family_id":"qf-density13-integer-opposite-difference","seed":202610132070,"semantic_template":"qf-density13-integer-opposite-difference/[7,-18]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=7, b=-18. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(-7, 18, 25, -25)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-7, 18, 25, -25)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0071","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28ab024daf0d88eb5a1caddc7b2b0c2e06d95ef4cecb12f17c6fab75ee4f8d43","family_id":"qf-density13-integer-opposite-difference","seed":202610132071,"semantic_template":"qf-density13-integer-opposite-difference/[8,-19]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=8, b=-19. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(-8, 19, 27, -27)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-8, 19, 27, -27)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0072","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e6f6f7e31d308cabda3fca6cebcbcd0c585ade6e01062d2b7f8867eec6658a6","family_id":"qf-density13-integer-opposite-difference","seed":202610132072,"semantic_template":"qf-density13-integer-opposite-difference/[9,-20]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=9, b=-20. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(-9, 20, 29, -29)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-9, 20, 29, -29)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0073","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3cd51fcac35a240513bbb37c797cfa4d352b730a227f76dbd4347ab220a4a74","family_id":"qf-density13-integer-opposite-difference","seed":202610132073,"semantic_template":"qf-density13-integer-opposite-difference/[10,-21]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=10, b=-21. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(-10, 21, 31, -31)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-10, 21, 31, -31)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0074","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41fc9e31d0eb64eb886fbc2863b9b6618f78a8100e470ea8ff766ae0a6d29770","family_id":"qf-density13-integer-opposite-difference","seed":202610132074,"semantic_template":"qf-density13-integer-opposite-difference/[11,-22]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=11, b=-22. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(-11, 22, 33, -33)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-11, 22, 33, -33)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0075","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71cedd86b321d3902cea08c21a2d936a1c93f87faf0c05cf0c50700d0e133e75","family_id":"qf-density13-integer-opposite-difference","seed":202610132075,"semantic_template":"qf-density13-integer-opposite-difference/[12,-23]"},"fields":{"title":"Ordered Opposites and Two Difference Directions","stem":"Integers a=12, b=-23. Give (−a, −b, a−b, b−a), retaining signs and direction.","answer":{"kind":"expression","value":"(-12, 23, 35, -35)","explanation":"Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-12, 23, 35, -35)."},"skills":["Ordered Opposites and Two Difference Directions"],"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-20261009-advance-20261013-r02-integer-0076","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6a4b06719bd0a3e2ecb8d71255bc18fa5095fd59a8a4fdb38937b9c34c70aaa","family_id":"qf-density13-integer-rational-grid-split","seed":202610132076,"semantic_template":"qf-density13-integer-rational-grid-split/[\"0/1\",\"-2/1\",\"1/2\",\"-1/3\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[0/1, -2/1, 1/2, 1/-3] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [0, -2], [1/2, -1/3])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [0, -2], [1/2, -1/3])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0077","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d32679c196281e3c4619f5dda34a2d4011e875b4b267da8d85043207dd12b71","family_id":"qf-density13-integer-rational-grid-split","seed":202610132077,"semantic_template":"qf-density13-integer-rational-grid-split/[\"-1/1\",\"3/1\",\"-2/1\",\"5/2\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[2/-2, 3/1, 4/-2, 5/2] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, Yes, No], [-1, 3, -2], [5/2])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, Yes, No], [-1, 3, -2], [5/2])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0078","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c614ded9579061dbfd16237b5ca9c58c5cb51eb2276c633ab5d33f51330df05","family_id":"qf-density13-integer-rational-grid-split","seed":202610132078,"semantic_template":"qf-density13-integer-rational-grid-split/[\"-10/1\",\"-2/1\",\"3/7\",\"4/9\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[-10/1, 4/-2, 3/7, -4/-9] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [-10, -2], [3/7, 4/9])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-10, -2], [3/7, 4/9])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0079","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e30a62ae446e1d62b91ca267eb0111f4821d2c25a3a8fa9206f50c7e7eec121","family_id":"qf-density13-integer-rational-grid-split","seed":202610132079,"semantic_template":"qf-density13-integer-rational-grid-split/[\"-9/1\",\"-3/1\",\"4/9\",\"5/11\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[-9/1, 6/-2, 4/9, -5/-11] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [-9, -3], [4/9, 5/11])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-9, -3], [4/9, 5/11])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0080","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b4803a52e87a7e8d64267aac5e46dc2f20cb41cec65dc7f2d64c87106b6106f","family_id":"qf-density13-integer-rational-grid-split","seed":202610132080,"semantic_template":"qf-density13-integer-rational-grid-split/[\"-8/1\",\"-4/1\",\"5/11\",\"6/13\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[-8/1, 8/-2, 5/11, -6/-13] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [-8, -4], [5/11, 6/13])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-8, -4], [5/11, 6/13])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0081","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9c521b37aaddf172dac9e9d84b94cac9aa1ac0765c9dee30acf0c85b4408625","family_id":"qf-density13-integer-rational-grid-split","seed":202610132081,"semantic_template":"qf-density13-integer-rational-grid-split/[\"-7/1\",\"-5/1\",\"6/13\",\"7/15\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[-7/1, 10/-2, 6/13, -7/-15] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [-7, -5], [6/13, 7/15])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-7, -5], [6/13, 7/15])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0082","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9a6d4b24832993999d5e85342640f0f8ff6bc58f478630cca0bce7f6d61d4de","family_id":"qf-density13-integer-rational-grid-split","seed":202610132082,"semantic_template":"qf-density13-integer-rational-grid-split/[\"-6/1\",\"-6/1\",\"7/15\",\"8/17\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[-6/1, 12/-2, 7/15, -8/-17] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [-6, -6], [7/15, 8/17])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-6, -6], [7/15, 8/17])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0083","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e53068d6bf2914b55d37c0124815d165def77a4c28bc4f6bc060bf7615184b9","family_id":"qf-density13-integer-rational-grid-split","seed":202610132083,"semantic_template":"qf-density13-integer-rational-grid-split/[\"-5/1\",\"-7/1\",\"8/17\",\"9/19\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[-5/1, 14/-2, 8/17, -9/-19] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [-5, -7], [8/17, 9/19])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-5, -7], [8/17, 9/19])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0084","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69e7dfed740193f24e3ab273b7d6056195f54fa84a6f482f19e572ebc8e5e393","family_id":"qf-density13-integer-rational-grid-split","seed":202610132084,"semantic_template":"qf-density13-integer-rational-grid-split/[\"-4/1\",\"-8/1\",\"9/19\",\"10/21\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[-4/1, 16/-2, 9/19, -10/-21] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [-4, -8], [9/19, 10/21])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-4, -8], [9/19, 10/21])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0085","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9527cf557afb73dcbaed349e284dd1cba095f7dd0199c5be5e564a9c2456e42c","family_id":"qf-density13-integer-rational-grid-split","seed":202610132085,"semantic_template":"qf-density13-integer-rational-grid-split/[\"-3/1\",\"-9/1\",\"10/21\",\"11/23\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[-3/1, 18/-2, 10/21, -11/-23] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [-3, -9], [10/21, 11/23])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-3, -9], [10/21, 11/23])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0086","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"197a54fd90a86ee5dad0ef83da7b4cec9dcf3063ce8bb66b24f5821dc4706ea1","family_id":"qf-density13-integer-rational-grid-split","seed":202610132086,"semantic_template":"qf-density13-integer-rational-grid-split/[\"-2/1\",\"-10/1\",\"11/23\",\"12/25\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[-2/1, 20/-2, 11/23, -12/-25] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [-2, -10], [11/23, 12/25])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-2, -10], [11/23, 12/25])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0087","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"590549aa64d2c13b250f2f5d7574b9ca5178907fc48c21bee0bc323e7217feee","family_id":"qf-density13-integer-rational-grid-split","seed":202610132087,"semantic_template":"qf-density13-integer-rational-grid-split/[\"-1/1\",\"-11/1\",\"12/25\",\"13/27\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[-1/1, 22/-2, 12/25, -13/-27] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [-1, -11], [12/25, 13/27])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-1, -11], [12/25, 13/27])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0088","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9b1599c3f394beafaa6c3f12687ae2e10aca811cafc245b6aa84f88af7599ff","family_id":"qf-density13-integer-rational-grid-split","seed":202610132088,"semantic_template":"qf-density13-integer-rational-grid-split/[\"0/1\",\"-12/1\",\"13/27\",\"14/29\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[0/1, 24/-2, 13/27, -14/-29] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [0, -12], [13/27, 14/29])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [0, -12], [13/27, 14/29])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0089","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89064bc69c922a3a851f457bb59febc67cb6264e389ded9104639b5b0f6caf91","family_id":"qf-density13-integer-rational-grid-split","seed":202610132089,"semantic_template":"qf-density13-integer-rational-grid-split/[\"1/1\",\"-13/1\",\"14/29\",\"15/31\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[1/1, 26/-2, 14/29, -15/-31] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [1, -13], [14/29, 15/31])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [1, -13], [14/29, 15/31])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0090","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d6af8b09f105955cf21abc8fe89a21b179d88c2e02e11004c75ccf0fe22b4d4","family_id":"qf-density13-integer-rational-grid-split","seed":202610132090,"semantic_template":"qf-density13-integer-rational-grid-split/[\"2/1\",\"-14/1\",\"15/31\",\"16/33\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[2/1, 28/-2, 15/31, -16/-33] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [2, -14], [15/31, 16/33])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [2, -14], [15/31, 16/33])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0091","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cad5104cfb0bbcbe440a4d6a968077259bd66b2e5647faa008a9551ce5917f40","family_id":"qf-density13-integer-rational-grid-split","seed":202610132091,"semantic_template":"qf-density13-integer-rational-grid-split/[\"3/1\",\"-15/1\",\"16/33\",\"17/35\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[3/1, 30/-2, 16/33, -17/-35] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [3, -15], [16/33, 17/35])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [3, -15], [16/33, 17/35])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0092","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fda4e590e3bf7871270eb665fdc27ca223c8a42a6cac5e6983279884052ee97a","family_id":"qf-density13-integer-rational-grid-split","seed":202610132092,"semantic_template":"qf-density13-integer-rational-grid-split/[\"4/1\",\"-16/1\",\"17/35\",\"18/37\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[4/1, 32/-2, 17/35, -18/-37] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [4, -16], [17/35, 18/37])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [4, -16], [17/35, 18/37])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0093","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60d1ff20cb3c243b508dd1bce0b2034706c063cbc10849c28d665c0aeceac979","family_id":"qf-density13-integer-rational-grid-split","seed":202610132093,"semantic_template":"qf-density13-integer-rational-grid-split/[\"5/1\",\"-17/1\",\"18/37\",\"19/39\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[5/1, 34/-2, 18/37, -19/-39] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [5, -17], [18/37, 19/39])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [5, -17], [18/37, 19/39])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0094","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32dc36a8575f04c56edfad6166242f0c4b925b5f4f70746b0657036cba752c35","family_id":"qf-density13-integer-rational-grid-split","seed":202610132094,"semantic_template":"qf-density13-integer-rational-grid-split/[\"6/1\",\"-18/1\",\"19/39\",\"20/41\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[6/1, 36/-2, 19/39, -20/-41] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [6, -18], [19/39, 20/41])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [6, -18], [19/39, 20/41])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0095","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ea2991ba6b534283e02be40cc004da89211e24916ab86e4e35d17c59f8b039c","family_id":"qf-density13-integer-rational-grid-split","seed":202610132095,"semantic_template":"qf-density13-integer-rational-grid-split/[\"7/1\",\"-19/1\",\"20/41\",\"21/43\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[7/1, 38/-2, 20/41, -21/-43] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [7, -19], [20/41, 21/43])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [7, -19], [20/41, 21/43])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0096","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64666995c69eb19cb932dc864cf38fd1e1b457d1fbd450b30e628f139ef39bab","family_id":"qf-density13-integer-rational-grid-split","seed":202610132096,"semantic_template":"qf-density13-integer-rational-grid-split/[\"8/1\",\"-20/1\",\"21/43\",\"22/45\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[8/1, 40/-2, 21/43, -22/-45] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [8, -20], [21/43, 22/45])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [8, -20], [21/43, 22/45])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0097","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be9f370afdb9de0cd58cb4c46c2a17066d068451b55bce243a72175b4013f9a1","family_id":"qf-density13-integer-rational-grid-split","seed":202610132097,"semantic_template":"qf-density13-integer-rational-grid-split/[\"9/1\",\"-21/1\",\"22/45\",\"23/47\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[9/1, 42/-2, 22/45, -23/-47] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [9, -21], [22/45, 23/47])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [9, -21], [22/45, 23/47])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0098","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14236ba74a48645ac3f8716be5ea019d40438d2a388306e66731c1e5d28ab84e","family_id":"qf-density13-integer-rational-grid-split","seed":202610132098,"semantic_template":"qf-density13-integer-rational-grid-split/[\"10/1\",\"-22/1\",\"23/47\",\"24/49\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[10/1, 44/-2, 23/47, -24/-49] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [10, -22], [23/47, 24/49])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [10, -22], [23/47, 24/49])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0099","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53bab5ead80de323f607e4ac74c69cca93c51c5393b3f9696cce7ced42438e4a","family_id":"qf-density13-integer-rational-grid-split","seed":202610132099,"semantic_template":"qf-density13-integer-rational-grid-split/[\"11/1\",\"-23/1\",\"24/49\",\"25/51\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[11/1, 46/-2, 24/49, -25/-51] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [11, -23], [24/49, 25/51])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [11, -23], [24/49, 25/51])."},"skills":["Rational Candidates Split by Integer 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-20261009-advance-20261013-r02-integer-0100","batch_id":"batch-20261009-advance-20261013-r02-integer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7dc3f3bba803af12a2fe45cbf1167332759363df4a479f40c08e2258d79d7278","family_id":"qf-density13-integer-rational-grid-split","seed":202610132100,"semantic_template":"qf-density13-integer-rational-grid-split/[\"12/1\",\"-24/1\",\"25/51\",\"26/53\"]"},"fields":{"title":"Rational Candidates Split by Integer Eligibility","stem":"Ordered representations=[12/1, 48/-2, 25/51, -26/-53] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], [12, -24], [25/51, 26/53])","explanation":"Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [12, -24], [25/51, 26/53])."},"skills":["Rational Candidates Split by Integer 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."}}