{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0001","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba307aaadfd397684db0e06fc777a6c14bf050ef64852e7559dc771ebf5651e1","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145001,"semantic_template":"qf-density14-factorial-adjacent-ratio/0"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=0. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(1, 1, 1)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (1, 1, 1)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0002","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"427867ef129122fccccd98c5cc340072d057c929332204764f40d5b984007f03","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145002,"semantic_template":"qf-density14-factorial-adjacent-ratio/1"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=1. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(1, 2, 2)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (1, 2, 2)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0003","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fadcc885c14c437504c2898194b7f1efc292307e04517523f6209e321a0e18a8","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145003,"semantic_template":"qf-density14-factorial-adjacent-ratio/2"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=2. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(2, 6, 3)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (2, 6, 3)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0004","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"853289f23a075328c16b65c98c14b1d8c3f92dcc0e79529a539086699a242717","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145004,"semantic_template":"qf-density14-factorial-adjacent-ratio/3"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=3. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(6, 24, 4)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (6, 24, 4)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0005","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"351a08b5e4380cacec5bbf83b743bd67c32b71ad67bed970a95f5b29f512662d","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145005,"semantic_template":"qf-density14-factorial-adjacent-ratio/4"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=4. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(24, 120, 5)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (24, 120, 5)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0006","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69b863ae55023429e843878997328d172a3b3ff21e5fa655769f2f23e41c8a85","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145006,"semantic_template":"qf-density14-factorial-adjacent-ratio/5"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=5. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(120, 720, 6)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (120, 720, 6)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0007","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"003f8eb6c69b86324339ace1da30f740e43ed03bc5c0bc78e9d03cb274383900","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145007,"semantic_template":"qf-density14-factorial-adjacent-ratio/6"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=6. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(720, 5040, 7)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (720, 5040, 7)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0008","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c36b91c5d1448377042bf56b91371f604e87c31c09f4ec6348098f91899a4425","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145008,"semantic_template":"qf-density14-factorial-adjacent-ratio/7"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=7. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(5040, 40320, 8)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (5040, 40320, 8)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0009","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fae287e31367980377779b18c0dcf77427fe79613c599eb130b7f680bec1a55","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145009,"semantic_template":"qf-density14-factorial-adjacent-ratio/8"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=8. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(40320, 362880, 9)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (40320, 362880, 9)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0010","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e318ee5ae71c21be1d85aaf26f2acffa1db9bb830456ec130cf933bb55eae07","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145010,"semantic_template":"qf-density14-factorial-adjacent-ratio/9"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=9. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(362880, 3628800, 10)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (362880, 3628800, 10)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0011","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"594678a684664186e6eb0825ea5280396042f87c4d1625b0da71b578cddef815","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145011,"semantic_template":"qf-density14-factorial-adjacent-ratio/10"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=10. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(3628800, 39916800, 11)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (3628800, 39916800, 11)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0012","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e549cfe38ef87044659dc3744e2ff5bc9c758d7ff5db10829f569a9fa863a420","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145012,"semantic_template":"qf-density14-factorial-adjacent-ratio/11"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=11. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(39916800, 479001600, 12)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (39916800, 479001600, 12)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0013","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7b84d1775364474a855792a632233b6e3f02715d02b3de0dedd8c646eda29fb","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145013,"semantic_template":"qf-density14-factorial-adjacent-ratio/12"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=12. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(479001600, 6227020800, 13)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (479001600, 6227020800, 13)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0014","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b0fde5e42e5205d46760a2bfc6633a81718bff3dd332d1b647cf089c8e70fb3","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145014,"semantic_template":"qf-density14-factorial-adjacent-ratio/13"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=13. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(6227020800, 87178291200, 14)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (6227020800, 87178291200, 14)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0015","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a70f8a21992fb9d5dd95d0a8caa457373bcea6f79a10add45a69bb9a5d11590","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145015,"semantic_template":"qf-density14-factorial-adjacent-ratio/14"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=14. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(87178291200, 1307674368000, 15)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (87178291200, 1307674368000, 15)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0016","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e01e59060f3e0d4f13029209f4aad3c3505aa3318431777551f989ead4246bf","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145016,"semantic_template":"qf-density14-factorial-adjacent-ratio/15"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=15. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(1307674368000, 20922789888000, 16)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (1307674368000, 20922789888000, 16)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0017","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9da896dd82bac1a38d0c654586985e283ebcdde810d0c488782221be8dd9fe26","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145017,"semantic_template":"qf-density14-factorial-adjacent-ratio/16"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=16. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(20922789888000, 355687428096000, 17)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (20922789888000, 355687428096000, 17)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0018","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c97a334ec8b193ee9c86d7f284eca05f903102f87734053d44e705194916b46c","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145018,"semantic_template":"qf-density14-factorial-adjacent-ratio/17"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=17. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(355687428096000, 6402373705728000, 18)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (355687428096000, 6402373705728000, 18)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0019","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad4bc801c6940604ded706024a6f9f2a1a4d8569b88db80836551550322595f5","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145019,"semantic_template":"qf-density14-factorial-adjacent-ratio/18"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=18. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(6402373705728000, 121645100408832000, 19)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (6402373705728000, 121645100408832000, 19)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0020","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"897fa01e963f6d59179a527ff87460871cce470f71e747251f3a3038eb78e837","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145020,"semantic_template":"qf-density14-factorial-adjacent-ratio/19"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=19. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(121645100408832000, 2432902008176640000, 20)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (121645100408832000, 2432902008176640000, 20)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0021","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e3bb0bd9bec6a8c79553bb41e8a10b8b552ad5a16150171ef51f11bbefb3460","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145021,"semantic_template":"qf-density14-factorial-adjacent-ratio/20"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=20. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(2432902008176640000, 51090942171709440000, 21)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (2432902008176640000, 51090942171709440000, 21)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0022","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9c9483276882bfdc68c22f1d2627a9aa177f63d1309bdda5bcfcea445e4dbc7","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145022,"semantic_template":"qf-density14-factorial-adjacent-ratio/21"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=21. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(51090942171709440000, 1124000727777607680000, 22)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (51090942171709440000, 1124000727777607680000, 22)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0023","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6786a954a1df92082f96164ac9c5bf2d70418327cfe3dbad39b7d51c2be59e70","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145023,"semantic_template":"qf-density14-factorial-adjacent-ratio/22"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=22. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(1124000727777607680000, 25852016738884976640000, 23)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (1124000727777607680000, 25852016738884976640000, 23)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0024","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1232a85f269b56a36a28ac4ea03444777d992c191c1ce9727ea3a292c236540e","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145024,"semantic_template":"qf-density14-factorial-adjacent-ratio/23"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=23. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(25852016738884976640000, 620448401733239439360000, 24)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (25852016738884976640000, 620448401733239439360000, 24)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0025","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2630901386902d5638f223ece54943ee7e3f6d1679796231e6b69a842fd93ff5","family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145025,"semantic_template":"qf-density14-factorial-adjacent-ratio/24"},"fields":{"title":"Adjacent Exact Factorials and Their Ratio","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=24. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).","answer":{"kind":"expression","value":"(620448401733239439360000, 15511210043330985984000000, 25)","explanation":"Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (620448401733239439360000, 15511210043330985984000000, 25)."},"skills":["Adjacent Exact Factorials and Their Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0026","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"737252ff00aaa55a8af9037e6063ca3e06ba68bffe166975c4d7bb99550b6f34","family_id":"qf-density14-factorial-window-product","seed":202610145026,"semantic_template":"qf-density14-factorial-window-product/[\"empty_window\"]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=0≤hi=0≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(1, [], Yes)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (1, [], Yes)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0027","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c4c379862795da47ad8585098cc0cce0eac1938594945568ce1b65ece0a4122","family_id":"qf-density14-factorial-window-product","seed":202610145027,"semantic_template":"qf-density14-factorial-window-product/[0,1]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=0≤hi=1≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(1, [1], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (1, [1], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0028","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c2c04075c652d76d53b3d8c66eb0e660cc0a01ee84a47bb2cf5ea79fd253743","family_id":"qf-density14-factorial-window-product","seed":202610145028,"semantic_template":"qf-density14-factorial-window-product/[0,20]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=0≤hi=20≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(2432902008176640000, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (2432902008176640000, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0029","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb2858cd9fb38793f2f6fde51e092d688079e269aab20f21dc0a1d187002061b","family_id":"qf-density14-factorial-window-product","seed":202610145029,"semantic_template":"qf-density14-factorial-window-product/[1,3]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=1≤hi=3≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(6, [2, 3], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (6, [2, 3], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0030","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b85c904307f5900ecfeae19974ae90df399c1e2bf66bae5ccd601786783d7cb","family_id":"qf-density14-factorial-window-product","seed":202610145030,"semantic_template":"qf-density14-factorial-window-product/[2,5]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=2≤hi=5≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(60, [3, 4, 5], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (60, [3, 4, 5], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0031","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab53ac8fefde4971a556fdf17dae15178719e19d2595993c0f5d2a94cd4aa177","family_id":"qf-density14-factorial-window-product","seed":202610145031,"semantic_template":"qf-density14-factorial-window-product/[3,7]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=3≤hi=7≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(840, [4, 5, 6, 7], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (840, [4, 5, 6, 7], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0032","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3718082881861711860d24f09455c12f212b2009f2585103fb3024bf26b2d7b7","family_id":"qf-density14-factorial-window-product","seed":202610145032,"semantic_template":"qf-density14-factorial-window-product/[4,6]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=4≤hi=6≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(30, [5, 6], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (30, [5, 6], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0033","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a677e8b0a72d79e291be6963b8d6368a463d95801c5fd8fd7cccf3e081dd7de","family_id":"qf-density14-factorial-window-product","seed":202610145033,"semantic_template":"qf-density14-factorial-window-product/[5,8]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=5≤hi=8≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(336, [6, 7, 8], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (336, [6, 7, 8], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0034","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4937401ea2beaceef7839c53cb81ce3992b5891e2df937184f15c67de27f311","family_id":"qf-density14-factorial-window-product","seed":202610145034,"semantic_template":"qf-density14-factorial-window-product/[6,10]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=6≤hi=10≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(5040, [7, 8, 9, 10], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (5040, [7, 8, 9, 10], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0035","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46704c07bbb760a57691ecc99bc5e7c606459a65054b4c5ad0233664b6e030aa","family_id":"qf-density14-factorial-window-product","seed":202610145035,"semantic_template":"qf-density14-factorial-window-product/[7,9]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=7≤hi=9≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(72, [8, 9], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (72, [8, 9], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0036","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7c66452d6b744bb1e3979150b0fa24a7f76ee9e52cb0cec68255018eade86ac","family_id":"qf-density14-factorial-window-product","seed":202610145036,"semantic_template":"qf-density14-factorial-window-product/[8,11]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=8≤hi=11≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(990, [9, 10, 11], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (990, [9, 10, 11], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0037","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6eabcd5290c55aa7173e571e621977c168894d01f9655b7e130717d1995c1407","family_id":"qf-density14-factorial-window-product","seed":202610145037,"semantic_template":"qf-density14-factorial-window-product/[9,13]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=9≤hi=13≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(17160, [10, 11, 12, 13], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (17160, [10, 11, 12, 13], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0038","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38b48a694c6b0cfc232e3157346d5e6864033ff43ebcf4d6bc9b690f56b62c36","family_id":"qf-density14-factorial-window-product","seed":202610145038,"semantic_template":"qf-density14-factorial-window-product/[10,12]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=10≤hi=12≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(132, [11, 12], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (132, [11, 12], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0039","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d4de0660b3f478b58f97c2b28e96510eb667c199dc9041bc2a493375fdbeea7","family_id":"qf-density14-factorial-window-product","seed":202610145039,"semantic_template":"qf-density14-factorial-window-product/[11,14]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=11≤hi=14≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(2184, [12, 13, 14], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (2184, [12, 13, 14], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0040","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d23c300b3163606157979b23824c4c5302dd97d71c3098b4955d1f7f27fd2a42","family_id":"qf-density14-factorial-window-product","seed":202610145040,"semantic_template":"qf-density14-factorial-window-product/[12,16]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=12≤hi=16≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(43680, [13, 14, 15, 16], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (43680, [13, 14, 15, 16], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0041","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea75c6f83d1d4b816c3f5f57f2a290e7f53ea879b1a58241d9273bdb51badfab","family_id":"qf-density14-factorial-window-product","seed":202610145041,"semantic_template":"qf-density14-factorial-window-product/[13,15]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=13≤hi=15≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(210, [14, 15], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (210, [14, 15], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0042","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bedb43c096b0f111cca39d2d4a8b880aa3b92d6a9cafd7ca82aeee1e879b963","family_id":"qf-density14-factorial-window-product","seed":202610145042,"semantic_template":"qf-density14-factorial-window-product/[14,17]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=14≤hi=17≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(4080, [15, 16, 17], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (4080, [15, 16, 17], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0043","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c93f083dbc9cf324b5a0a977b716a6ef46871f1b3d695a64a933b5fc217987a8","family_id":"qf-density14-factorial-window-product","seed":202610145043,"semantic_template":"qf-density14-factorial-window-product/[15,19]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=15≤hi=19≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(93024, [16, 17, 18, 19], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (93024, [16, 17, 18, 19], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0044","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"853ae34f249bdafa350e2c90c5dd0c37fe5fabc98501225ffcf213891c6782d3","family_id":"qf-density14-factorial-window-product","seed":202610145044,"semantic_template":"qf-density14-factorial-window-product/[16,18]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=16≤hi=18≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(306, [17, 18], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (306, [17, 18], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0045","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9cbc097853871369d0532bb9e7404cd431f9d07d98375d132f5621fd7354094","family_id":"qf-density14-factorial-window-product","seed":202610145045,"semantic_template":"qf-density14-factorial-window-product/[17,20]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=17≤hi=20≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(6840, [18, 19, 20], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (6840, [18, 19, 20], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0046","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11b15a98a3bfb6c0aefc36303e34cd250642f1e192d5b71bd208d76e215dfbf7","family_id":"qf-density14-factorial-window-product","seed":202610145046,"semantic_template":"qf-density14-factorial-window-product/[18,22]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=18≤hi=22≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(175560, [19, 20, 21, 22], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (175560, [19, 20, 21, 22], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0047","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"492147c1879938fb3e674e152bba917e830b365fbe8530b6570109a1199a08f8","family_id":"qf-density14-factorial-window-product","seed":202610145047,"semantic_template":"qf-density14-factorial-window-product/[19,21]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=19≤hi=21≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(420, [20, 21], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (420, [20, 21], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0048","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9ed8444ee04892dd2a586177605f907e4da2c80d87fcf0c5d2bd65033a3dba4","family_id":"qf-density14-factorial-window-product","seed":202610145048,"semantic_template":"qf-density14-factorial-window-product/[20,23]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=20≤hi=23≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(10626, [21, 22, 23], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (10626, [21, 22, 23], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0049","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6ff62d6dee33d1592ce2c24c7fed3e9b0a0dd6ce75ce9146e9c2bf5949e5512","family_id":"qf-density14-factorial-window-product","seed":202610145049,"semantic_template":"qf-density14-factorial-window-product/[21,25]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=21≤hi=25≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(303600, [22, 23, 24, 25], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (303600, [22, 23, 24, 25], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0050","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04cc63e1dad312515eca1a7793939a6cc307a3a908f9ea1257e4ceb61a1e5643","family_id":"qf-density14-factorial-window-product","seed":202610145050,"semantic_template":"qf-density14-factorial-window-product/[22,24]"},"fields":{"title":"Factorial Window Product and Empty-Window Eligibility","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=22≤hi=24≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).","answer":{"kind":"expression","value":"(552, [23, 24], No)","explanation":"Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (552, [23, 24], No)."},"skills":["Factorial Window Product and Empty-Window 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-20261010-advance-20261014-r05-factorial-0051","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac258e20d3d4f0ddd4cb4057a62bfd244714838efd6e56bb105fb4b815c6fd91","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145051,"semantic_template":"qf-density14-factorial-reported-successor-audit/[0,\"1\",\"1\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=0; candidate decimal integer strings u=1, v=1. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(1, 1, 0, 0, Yes)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (1, 1, 0, 0, Yes)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0052","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d204f9070306e59549c131c10de1d03c57bf7bd3644b5051654e57afcc5d31c","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145052,"semantic_template":"qf-density14-factorial-reported-successor-audit/[3,\"12\",\"48\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=3; candidate decimal integer strings u=12, v=48. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(6, 24, 6, 0, No)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (6, 24, 6, 0, No)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0053","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00c2a11d1e6471d28cd654b7132966ddcbb264d844eb37c1a74d2c2d5b023439","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145053,"semantic_template":"qf-density14-factorial-reported-successor-audit/[2,\"2\",\"7\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=2; candidate decimal integer strings u=2, v=7. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(2, 6, 0, 1, No)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (2, 6, 0, 1, No)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0054","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a32011bea8921b471b8f2f8067c1ca650cc6383984e7807ee0b6e1315f0b357c","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145054,"semantic_template":"qf-density14-factorial-reported-successor-audit/[3,\"6\",\"24\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=3; candidate decimal integer strings u=6, v=24. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(6, 24, 0, 0, Yes)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (6, 24, 0, 0, Yes)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0055","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31fec6af76e11a8d7e747ced0b0de00ec419992183549d8d3221c3b4e4a94f81","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145055,"semantic_template":"qf-density14-factorial-reported-successor-audit/[4,\"25\",\"125\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=4; candidate decimal integer strings u=25, v=125. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(24, 120, 1, 0, No)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (24, 120, 1, 0, No)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0056","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9ac2cfa2f64253bb2af0d41b9f96b1cb6db68b0c3b3483852919d91f898ea5f","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145056,"semantic_template":"qf-density14-factorial-reported-successor-audit/[5,\"120\",\"721\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=5; candidate decimal integer strings u=120, v=721. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(120, 720, 0, 1, No)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (120, 720, 0, 1, No)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0057","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9466d271d40fe704c97dc101f676f250808729311dc0321f003fb233e3b773d6","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145057,"semantic_template":"qf-density14-factorial-reported-successor-audit/[6,\"720\",\"5040\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=6; candidate decimal integer strings u=720, v=5040. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(720, 5040, 0, 0, Yes)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (720, 5040, 0, 0, Yes)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0058","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b5123ffb8e8f16b3f99826a64d5f24c0c1e87dddaa5bc60d566cd107b18b3d0","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145058,"semantic_template":"qf-density14-factorial-reported-successor-audit/[7,\"5041\",\"40328\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=7; candidate decimal integer strings u=5041, v=40328. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(5040, 40320, 1, 0, No)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (5040, 40320, 1, 0, No)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0059","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d31724f7052d1c22f18858b6909822f10682718cd4377329ee5d8a2c5465c6d8","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145059,"semantic_template":"qf-density14-factorial-reported-successor-audit/[8,\"40320\",\"362881\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=8; candidate decimal integer strings u=40320, v=362881. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(40320, 362880, 0, 1, No)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (40320, 362880, 0, 1, No)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0060","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51d7beeb20aab9912e3d3b8f02afe51d1f847e4d8c67dd7e49d6680ac0cb55af","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145060,"semantic_template":"qf-density14-factorial-reported-successor-audit/[9,\"362880\",\"3628800\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=9; candidate decimal integer strings u=362880, v=3628800. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(362880, 3628800, 0, 0, Yes)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (362880, 3628800, 0, 0, Yes)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0061","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05bf07071be3e0d4f7a820638afc7a7128ce9a586c9e625809ff2d5f3c41f64f","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145061,"semantic_template":"qf-density14-factorial-reported-successor-audit/[10,\"3628801\",\"39916811\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=10; candidate decimal integer strings u=3628801, v=39916811. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(3628800, 39916800, 1, 0, No)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (3628800, 39916800, 1, 0, No)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0062","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b23a2e0b4901685ec3bfc6df7595acbed37384489bda464ab5ad88e0e125949a","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145062,"semantic_template":"qf-density14-factorial-reported-successor-audit/[11,\"39916800\",\"479001601\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=11; candidate decimal integer strings u=39916800, v=479001601. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(39916800, 479001600, 0, 1, No)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (39916800, 479001600, 0, 1, No)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0063","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bc2e3c79fb145dd4a19c5b25063ffac4d59d40e2ddf91bcbed777840a8afdfb","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145063,"semantic_template":"qf-density14-factorial-reported-successor-audit/[12,\"479001600\",\"6227020800\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=12; candidate decimal integer strings u=479001600, v=6227020800. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(479001600, 6227020800, 0, 0, Yes)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (479001600, 6227020800, 0, 0, Yes)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0064","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"814242e4954217b3e3972379f17f211643a8f9abbc65cb70ad8a29bcc41c2292","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145064,"semantic_template":"qf-density14-factorial-reported-successor-audit/[13,\"6227020801\",\"87178291214\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=13; candidate decimal integer strings u=6227020801, v=87178291214. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(6227020800, 87178291200, 1, 0, No)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (6227020800, 87178291200, 1, 0, No)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0065","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43355fa5c653c5f549f3ab6b5b256d232fefb8b0daf8b45f452f542d3a8b2d54","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145065,"semantic_template":"qf-density14-factorial-reported-successor-audit/[14,\"87178291200\",\"1307674368001\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=14; candidate decimal integer strings u=87178291200, v=1307674368001. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(87178291200, 1307674368000, 0, 1, No)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (87178291200, 1307674368000, 0, 1, No)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0066","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f7b281c17fd32f4b03e750917fbb0139ed3f2163551f3c170a216af6bd16083","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145066,"semantic_template":"qf-density14-factorial-reported-successor-audit/[15,\"1307674368000\",\"20922789888000\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=15; candidate decimal integer strings u=1307674368000, v=20922789888000. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(1307674368000, 20922789888000, 0, 0, Yes)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (1307674368000, 20922789888000, 0, 0, Yes)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0067","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a57c472753441fc7f7c785d52581109e4c8a113db094abd6e14deb29fd0c91af","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145067,"semantic_template":"qf-density14-factorial-reported-successor-audit/[16,\"20922789888001\",\"355687428096017\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=16; candidate decimal integer strings u=20922789888001, v=355687428096017. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(20922789888000, 355687428096000, 1, 0, No)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (20922789888000, 355687428096000, 1, 0, No)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0068","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b0ac6c4a8db3c790bec1f12a4269e317e0d40ef55caa28805767db98dc016cf","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145068,"semantic_template":"qf-density14-factorial-reported-successor-audit/[17,\"355687428096000\",\"6402373705728001\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=17; candidate decimal integer strings u=355687428096000, v=6402373705728001. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(355687428096000, 6402373705728000, 0, 1, No)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (355687428096000, 6402373705728000, 0, 1, No)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0069","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b15899c64de62892f632e68b9cc71d976de6ac4285b2171154a89f5a4e5aaf72","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145069,"semantic_template":"qf-density14-factorial-reported-successor-audit/[18,\"6402373705728000\",\"121645100408832000\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=18; candidate decimal integer strings u=6402373705728000, v=121645100408832000. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(6402373705728000, 121645100408832000, 0, 0, Yes)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (6402373705728000, 121645100408832000, 0, 0, Yes)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0070","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"590134b71c33efe4d48b15be9b7333507a57a6d50b96eb5218d0625d8399f83b","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145070,"semantic_template":"qf-density14-factorial-reported-successor-audit/[19,\"121645100408832001\",\"2432902008176640020\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=19; candidate decimal integer strings u=121645100408832001, v=2432902008176640020. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(121645100408832000, 2432902008176640000, 1, 0, No)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (121645100408832000, 2432902008176640000, 1, 0, No)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0071","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac3436f55abdd3179165ee4512881c8f9f43b6f379ee7d0d60c2b911c0c058b4","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145071,"semantic_template":"qf-density14-factorial-reported-successor-audit/[20,\"2432902008176640000\",\"51090942171709440001\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=20; candidate decimal integer strings u=2432902008176640000, v=51090942171709440001. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(2432902008176640000, 51090942171709440000, 0, 1, No)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (2432902008176640000, 51090942171709440000, 0, 1, No)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0072","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"738cc483b6a476f9a7ddeb0025d1ed7eb4eae293b060dce5e92a87355cb7ef8c","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145072,"semantic_template":"qf-density14-factorial-reported-successor-audit/[21,\"51090942171709440000\",\"1124000727777607680000\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=21; candidate decimal integer strings u=51090942171709440000, v=1124000727777607680000. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(51090942171709440000, 1124000727777607680000, 0, 0, Yes)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (51090942171709440000, 1124000727777607680000, 0, 0, Yes)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0073","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3d856752c58d74d608f455bcf3b3889ebe092a05670f615d7d88a784d2d75c7","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145073,"semantic_template":"qf-density14-factorial-reported-successor-audit/[22,\"1124000727777607680001\",\"25852016738884976640023\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=22; candidate decimal integer strings u=1124000727777607680001, v=25852016738884976640023. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(1124000727777607680000, 25852016738884976640000, 1, 0, No)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (1124000727777607680000, 25852016738884976640000, 1, 0, No)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0074","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"678ab55c17469583b030afec250b6f938da02c0d4ef7453695a5bdc0560d6047","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145074,"semantic_template":"qf-density14-factorial-reported-successor-audit/[23,\"25852016738884976640000\",\"620448401733239439360001\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=23; candidate decimal integer strings u=25852016738884976640000, v=620448401733239439360001. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(25852016738884976640000, 620448401733239439360000, 0, 1, No)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (25852016738884976640000, 620448401733239439360000, 0, 1, No)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0075","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffd763c17e6c9d2fdfdd3e7e4b2098f6abcd48f06c79515cddc82d48c2bfa604","family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145075,"semantic_template":"qf-density14-factorial-reported-successor-audit/[24,\"620448401733239439360000\",\"15511210043330985984000000\"]"},"fields":{"title":"Reported Factorials versus Exact Values and Recurrence","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=24; candidate decimal integer strings u=620448401733239439360000, v=15511210043330985984000000. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).","answer":{"kind":"expression","value":"(620448401733239439360000, 15511210043330985984000000, 0, 0, Yes)","explanation":"Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (620448401733239439360000, 15511210043330985984000000, 0, 0, Yes)."},"skills":["Reported Factorials versus Exact Values and Recurrence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0076","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2c43c3e7852f77ec0cec5166cb38a6431f85cc9db1bb13f66a4603a7d1b6a23","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145076,"semantic_template":"qf-density14-factorial-single-factor-omission/[1,1]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=1, k=1∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(1, 1, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (1, 1, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0077","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5af6be878e8875b2886035fc16005a7d1a1adee4d85eb4970b70e864d84c2aac","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145077,"semantic_template":"qf-density14-factorial-single-factor-omission/[19,19]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=19, k=19∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(121645100408832000, 6402373705728000, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (121645100408832000, 6402373705728000, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0078","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d2827405f2f6068c178d6329b0d115eb7cd48005836c0434b7ed438592d150f","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145078,"semantic_template":"qf-density14-factorial-single-factor-omission/[4,3]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=4, k=3∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(24, 8, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (24, 8, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0079","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6c02d3f0107978b72733f2e24e7d88a27a63953c7bd6970feb73fe489dd043f","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145079,"semantic_template":"qf-density14-factorial-single-factor-omission/[5,4]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=5, k=4∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(120, 30, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (120, 30, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0080","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"335f8a2cc77dc05f19336331f8f8db49998bb135ca3f8ab3839fad263b35941c","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145080,"semantic_template":"qf-density14-factorial-single-factor-omission/[6,5]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=6, k=5∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(720, 144, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (720, 144, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0081","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fdd51a5d702856be4d3af648fca43567dcce65b09afd0dd7799bbaab908b223","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145081,"semantic_template":"qf-density14-factorial-single-factor-omission/[7,6]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=7, k=6∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(5040, 840, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (5040, 840, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0082","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"febf3589050808ad28d21f4b72339620acec846dc53d32eaca80594d158a9008","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145082,"semantic_template":"qf-density14-factorial-single-factor-omission/[8,7]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=8, k=7∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(40320, 5760, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (40320, 5760, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0083","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb9bf12e0637b49d3c0bc4d2113da9e693e0c4cf9098bfee7057beeafd50c2e5","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145083,"semantic_template":"qf-density14-factorial-single-factor-omission/[9,8]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=9, k=8∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(362880, 45360, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (362880, 45360, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0084","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55fd85cef7e6e4fd2af7b8b2ba2db3597a74766758b863ad949783fb12c29c1b","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145084,"semantic_template":"qf-density14-factorial-single-factor-omission/[10,9]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=10, k=9∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(3628800, 403200, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (3628800, 403200, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0085","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f64df6c8da9f5aad9557b189b840a87e7ab37667529b33ee8d76bc12d843bec8","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145085,"semantic_template":"qf-density14-factorial-single-factor-omission/[11,10]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=11, k=10∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(39916800, 3991680, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (39916800, 3991680, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0086","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6562d8010a9d389609c7ae521d7422016d04b65ef46bcc0ba29ce75c0d367d43","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145086,"semantic_template":"qf-density14-factorial-single-factor-omission/[12,11]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=12, k=11∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(479001600, 43545600, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (479001600, 43545600, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0087","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f0ca7c97845560747dba7139541c635676cca08a4558637424ef79e16cc660c","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145087,"semantic_template":"qf-density14-factorial-single-factor-omission/[13,12]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=13, k=12∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(6227020800, 518918400, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (6227020800, 518918400, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0088","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64b1e120e5af0c3041a6de4b430a2f163e57e65ccee83cac0c4cb3c1f90bdf80","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145088,"semantic_template":"qf-density14-factorial-single-factor-omission/[14,13]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=14, k=13∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(87178291200, 6706022400, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (87178291200, 6706022400, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0089","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a71ba12ee960ea8a45d7ceb97ea67063510684bef50946ecec8965579b1a4698","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145089,"semantic_template":"qf-density14-factorial-single-factor-omission/[15,14]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=15, k=14∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(1307674368000, 93405312000, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (1307674368000, 93405312000, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0090","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77fb7ed973b031327787eda87a93f4ae8ffad65b436fbc99a512c742662da574","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145090,"semantic_template":"qf-density14-factorial-single-factor-omission/[16,15]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=16, k=15∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(20922789888000, 1394852659200, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (20922789888000, 1394852659200, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0091","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1135b6ff8a5e1c565b894b01813c07a71ad24be7a7a043c5c8f9ce38dd3b349","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145091,"semantic_template":"qf-density14-factorial-single-factor-omission/[17,16]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=17, k=16∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(355687428096000, 22230464256000, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (355687428096000, 22230464256000, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0092","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4b4be0f52e7deff0b4ad6ac4a2f4cd02d3fb0169245973039c4a1aa34be53ea","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145092,"semantic_template":"qf-density14-factorial-single-factor-omission/[18,17]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=18, k=17∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(6402373705728000, 376610217984000, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (6402373705728000, 376610217984000, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0093","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6366f59c3fe044ab9352058ed21add0fb9d753bcd7aaa5e033bfbdabe28a2fc3","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145093,"semantic_template":"qf-density14-factorial-single-factor-omission/[19,18]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=19, k=18∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(121645100408832000, 6758061133824000, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (121645100408832000, 6758061133824000, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0094","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29d6ed80816248ce887fe5e2d0ee5315b6ca416170ea73b2c70a93747f02897f","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145094,"semantic_template":"qf-density14-factorial-single-factor-omission/[20,19]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=20, k=19∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(2432902008176640000, 128047474114560000, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (2432902008176640000, 128047474114560000, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0095","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57a9d222fde4147b382e4f65f9e5d32f500a673e46ce15991607dc1efeefc6b3","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145095,"semantic_template":"qf-density14-factorial-single-factor-omission/[21,20]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=21, k=20∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(51090942171709440000, 2554547108585472000, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (51090942171709440000, 2554547108585472000, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0096","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31508aee4c512d991263571855001afd1b4582f5a29c2fa8cb0f171364dad1ba","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145096,"semantic_template":"qf-density14-factorial-single-factor-omission/[22,21]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=22, k=21∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(1124000727777607680000, 53523844179886080000, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (1124000727777607680000, 53523844179886080000, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0097","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c71592a537c13baa603cc6910a82ebcf3dd472bece3aed81bc3486bef93e22d","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145097,"semantic_template":"qf-density14-factorial-single-factor-omission/[23,22]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=23, k=22∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(25852016738884976640000, 1175091669949317120000, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (25852016738884976640000, 1175091669949317120000, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0098","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc5b3c5f26832acfc9eb4176201df923808b53f8a130c82d51e877d9d84e69f2","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145098,"semantic_template":"qf-density14-factorial-single-factor-omission/[24,23]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=24, k=23∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(620448401733239439360000, 26976017466662584320000, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (620448401733239439360000, 26976017466662584320000, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0099","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7d37b0f055f27306b2fd8f764a76ce0797facde26460b9352dad18c7c2b53eb","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145099,"semantic_template":"qf-density14-factorial-single-factor-omission/[25,24]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=25, k=24∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(15511210043330985984000000, 646300418472124416000000, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (15511210043330985984000000, 646300418472124416000000, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r05-factorial-0100","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efbb88fa951e8a4ad9c9262f29d8ed2de7bbd904c3b0717b5d91578cc904b269","family_id":"qf-density14-factorial-single-factor-omission","seed":202610145100,"semantic_template":"qf-density14-factorial-single-factor-omission/[26,25]"},"fields":{"title":"Omitting One Existing Factor and Multiplying Back","stem":"N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=26, k=25∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).","answer":{"kind":"expression","value":"(403291461126605635584000000, 16131658445064225423360000, 0)","explanation":"Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (403291461126605635584000000, 16131658445064225423360000, 0)."},"skills":["Omitting One Existing Factor and Multiplying Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}