{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0001","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a88441b3e2b988b7b3c4d9542fb1d3f2ecb7f0899c73f065b0b30ddac2cd8dde","family_id":"qf-factorial-exact-evaluation","seed":20260927,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":0,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=0; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"1","explanation":"0!=1, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0002","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0860f99bffe6cad3df9cb42ef976361f619c1964789e830e9d81077cbd02ea89","family_id":"qf-factorial-exact-evaluation","seed":20260928,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":1,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=1; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"1","explanation":"1!=1, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0003","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"068133ae7c3beb03a9ac3ac44bf026541711fc09ea15cba3dd5174380bbd386c","family_id":"qf-factorial-exact-evaluation","seed":20260929,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":2,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=2; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"2","explanation":"2!=2, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0004","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc0c17347498c0a483563b58400ec297b157ec4e9092b21f6cdbaf69d4ea1f08","family_id":"qf-factorial-exact-evaluation","seed":20260930,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":3,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=3; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"6","explanation":"3!=6, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0005","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fe91c19057e360632cf1e53cf1c0ba0875b35a98bd142546b84f1242dd796fc","family_id":"qf-factorial-exact-evaluation","seed":20260931,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":4,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=4; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"24","explanation":"4!=24, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0006","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e87f48738dbbf2366bc18bc6997c9216cd9c248a391eb160041ef3adb4a3e52","family_id":"qf-factorial-exact-evaluation","seed":20260932,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":5,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=5; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"120","explanation":"5!=120, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0007","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac2eb40c372c2805810890149c5297947c9978d31ac3943d76ca98a275f28e44","family_id":"qf-factorial-exact-evaluation","seed":20260933,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":6,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=6; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"720","explanation":"6!=720, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0008","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52ff01397d285d7f6013cdb4eca6a1e1cdd7882a996da8f92fb50faa94373ce0","family_id":"qf-factorial-exact-evaluation","seed":20260934,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":7,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=7; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"5040","explanation":"7!=5040, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0009","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec9347bf1df25a209d7b9b703b0f4e9eecd104f83bdc307c9bb96ddc55b10c31","family_id":"qf-factorial-exact-evaluation","seed":20260935,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":8,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=8; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"40320","explanation":"8!=40320, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0010","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cbb254ba36626f1d2e80dc0afe3d31e7fc1dbe765fda9ab30b394b22df11e63","family_id":"qf-factorial-exact-evaluation","seed":20260936,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":9,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=9; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"362880","explanation":"9!=362880, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0011","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb61a5bc29f5df42b29f11804e2731b865c7ceee495bd52892d38da8e8c9e9f0","family_id":"qf-factorial-exact-evaluation","seed":20260937,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":10,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=10; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"3628800","explanation":"10!=3628800, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0012","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf24bb5949d1fca2bee2ebed0acac139f2bd46d7b7a6bb48ee9e5665bc0081fc","family_id":"qf-factorial-exact-evaluation","seed":20260938,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":11,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=11; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"39916800","explanation":"11!=39916800, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0013","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1e6d515e08f49acf1e2678953e69e2cdeb4088be94dba94b6a2de7e344ec735","family_id":"qf-factorial-exact-evaluation","seed":20260939,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":12,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=12; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"479001600","explanation":"12!=479001600, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0014","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"388665688b73774ffdc4eef4a49eee54baae033b77175520c59cbbd874b34999","family_id":"qf-factorial-exact-evaluation","seed":20260940,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":13,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=13; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"6227020800","explanation":"13!=6227020800, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0015","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d39f3741b900c112b2df1f06f9223edcadd8a4fcd7e13440b11498e1521ffa53","family_id":"qf-factorial-exact-evaluation","seed":20260941,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":14,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=14; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"87178291200","explanation":"14!=87178291200, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0016","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af140eda8049ee810e94458bff1bad24382a166381054a2a086811883fcd8830","family_id":"qf-factorial-exact-evaluation","seed":20260942,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":15,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=15; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"1307674368000","explanation":"15!=1307674368000, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0017","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cecfb9790a233db1debe039bad4d595631d6550a8a3ef9fa8c038eae02d8aed","family_id":"qf-factorial-exact-evaluation","seed":20260943,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":16,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=16; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"20922789888000","explanation":"16!=20922789888000, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0018","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37e8addf0b7d9fea8d54df1ae255710e9ef0325cce42eb9b27f803965019de0e","family_id":"qf-factorial-exact-evaluation","seed":20260944,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":17,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=17; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"355687428096000","explanation":"17!=355687428096000, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0019","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43c47c97d0d492e0fbec6dae340b3126380523a3e7d9d3ca302429ebe5982407","family_id":"qf-factorial-exact-evaluation","seed":20260945,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":18,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=18; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"6402373705728000","explanation":"18!=6402373705728000, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0020","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f3f7b7242eb56fbf001cf8cae5829d24d886efaaa19cb61e282c6438e216617","family_id":"qf-factorial-exact-evaluation","seed":20260946,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":19,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=19; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"121645100408832000","explanation":"19!=121645100408832000, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0021","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f418c77d7d6c833f55dcd2ecd1a306bb4a116f7f66c55dcd210a2b7f2c28cbc","family_id":"qf-factorial-exact-evaluation","seed":20260947,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":20,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=20; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"2432902008176640000","explanation":"20!=2432902008176640000, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0022","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2010c625fada878fad9edeb73ad143d9b962f9b0258bc52b34dff6383c9da82","family_id":"qf-factorial-exact-evaluation","seed":20260948,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":21,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=21; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"51090942171709440000","explanation":"21!=51090942171709440000, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0023","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0959fb4d05c4391f42edb424abb2667c0f73f3a7082ffcec628fc2e8cdcf40e3","family_id":"qf-factorial-exact-evaluation","seed":20260949,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":22,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=22; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"1124000727777607680000","explanation":"22!=1124000727777607680000, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0024","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"896f848713672f7a5632f53c08bf0eb00d4f51ac01e71e7e0c378b83601b9ffc","family_id":"qf-factorial-exact-evaluation","seed":20260950,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":23,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=23; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"25852016738884976640000","explanation":"23!=25852016738884976640000, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-exact-evaluation-0025","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a82064388d22fd9e37be3ad570359cdac1e94dcf87193c82f8af5605e719e526","family_id":"qf-factorial-exact-evaluation","seed":20260951,"semantic_template":"qf-factorial-exact-evaluation/{\"n\":24,\"operation\":\"exact_factorial\"}"},"fields":{"title":"Exact Factorial Value","stem":"Compute n! for the natural number n=24; give the answer as a decimal integer string.","answer":{"kind":"short_text","value":"620448401733239439360000","explanation":"24!=620448401733239439360000, computed exactly with arbitrary-precision integer arithmetic."},"skills":["factorial","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0001","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"112c22c3466904663b64440d32a645649accf60a8c84c1360b4a5378c7abd7e6","family_id":"qf-factorial-successor-recurrence","seed":20261027,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":0,\"n_factorial\":\"1\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 0!=1, use (n+1)!=(n+1)×n! to find 1!.","answer":{"kind":"short_text","value":"1","explanation":"1×1=1, so 1!=1."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0002","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b792dfb6b272c37aabd23aeeb262f06b1dc31e383a84c54b278b2b8111696c45","family_id":"qf-factorial-successor-recurrence","seed":20261028,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":1,\"n_factorial\":\"1\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 1!=1, use (n+1)!=(n+1)×n! to find 2!.","answer":{"kind":"short_text","value":"2","explanation":"2×1=2, so 2!=2."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0003","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b61b5a6e8eedb481c05829768d0d06c80aebd23eff2d13664be07e6b211b1ec9","family_id":"qf-factorial-successor-recurrence","seed":20261029,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":2,\"n_factorial\":\"2\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 2!=2, use (n+1)!=(n+1)×n! to find 3!.","answer":{"kind":"short_text","value":"6","explanation":"3×2=6, so 3!=6."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0004","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01890ea71c616255aebf558c7f5d3887269f1efb8b881cfb3a6a0d968a2e13a5","family_id":"qf-factorial-successor-recurrence","seed":20261030,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":3,\"n_factorial\":\"6\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 3!=6, use (n+1)!=(n+1)×n! to find 4!.","answer":{"kind":"short_text","value":"24","explanation":"4×6=24, so 4!=24."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0005","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"437a6bfa49c1d6f6f36a36fd0b80997d4a98acd27fc66d1f92f59a6c706a2237","family_id":"qf-factorial-successor-recurrence","seed":20261031,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":4,\"n_factorial\":\"24\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 4!=24, use (n+1)!=(n+1)×n! to find 5!.","answer":{"kind":"short_text","value":"120","explanation":"5×24=120, so 5!=120."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0006","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f3d8c4fda10c27efa97d1736a630d593948476936eb0e7cf3d460c298d56caa","family_id":"qf-factorial-successor-recurrence","seed":20261032,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":5,\"n_factorial\":\"120\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 5!=120, use (n+1)!=(n+1)×n! to find 6!.","answer":{"kind":"short_text","value":"720","explanation":"6×120=720, so 6!=720."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0007","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cab3cab60f5a397abe5aca17d11d6e2305f24c23265ed56bff2ffadc51ca72d4","family_id":"qf-factorial-successor-recurrence","seed":20261033,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":6,\"n_factorial\":\"720\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 6!=720, use (n+1)!=(n+1)×n! to find 7!.","answer":{"kind":"short_text","value":"5040","explanation":"7×720=5040, so 7!=5040."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0008","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d64d24a468e4540f3fec69ceded19bfe37ba9d7f6eb40207266f3c5b36e1893a","family_id":"qf-factorial-successor-recurrence","seed":20261034,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":7,\"n_factorial\":\"5040\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 7!=5040, use (n+1)!=(n+1)×n! to find 8!.","answer":{"kind":"short_text","value":"40320","explanation":"8×5040=40320, so 8!=40320."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0009","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c0e9ba4c5ffa20907e50f7f988f000f60a27aad9feebbce4ecf98b10fac6eb7","family_id":"qf-factorial-successor-recurrence","seed":20261035,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":8,\"n_factorial\":\"40320\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 8!=40320, use (n+1)!=(n+1)×n! to find 9!.","answer":{"kind":"short_text","value":"362880","explanation":"9×40320=362880, so 9!=362880."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0010","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8105487582d0b218272debe270bfd6be9e6017ba6c0652ff92172c3e8b0c8b3a","family_id":"qf-factorial-successor-recurrence","seed":20261036,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":9,\"n_factorial\":\"362880\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 9!=362880, use (n+1)!=(n+1)×n! to find 10!.","answer":{"kind":"short_text","value":"3628800","explanation":"10×362880=3628800, so 10!=3628800."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0011","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a60143841fca9cc3da4a76e50fc60326270d608637c1a8920c1020a8270a1622","family_id":"qf-factorial-successor-recurrence","seed":20261037,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":10,\"n_factorial\":\"3628800\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 10!=3628800, use (n+1)!=(n+1)×n! to find 11!.","answer":{"kind":"short_text","value":"39916800","explanation":"11×3628800=39916800, so 11!=39916800."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0012","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0022985433c2671662f9fd202959146c51cd17186216d4757d207447a1afb608","family_id":"qf-factorial-successor-recurrence","seed":20261038,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":11,\"n_factorial\":\"39916800\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 11!=39916800, use (n+1)!=(n+1)×n! to find 12!.","answer":{"kind":"short_text","value":"479001600","explanation":"12×39916800=479001600, so 12!=479001600."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0013","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e53ef668812ee260b073d9a8cb82b91d07a2549d73fecd7e3e803c2b8c349f7","family_id":"qf-factorial-successor-recurrence","seed":20261039,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":12,\"n_factorial\":\"479001600\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 12!=479001600, use (n+1)!=(n+1)×n! to find 13!.","answer":{"kind":"short_text","value":"6227020800","explanation":"13×479001600=6227020800, so 13!=6227020800."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0014","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e11de0d4f358c1077b03f573db2b7c40454b0086c3f303138b4673863b89c19","family_id":"qf-factorial-successor-recurrence","seed":20261040,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":13,\"n_factorial\":\"6227020800\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 13!=6227020800, use (n+1)!=(n+1)×n! to find 14!.","answer":{"kind":"short_text","value":"87178291200","explanation":"14×6227020800=87178291200, so 14!=87178291200."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0015","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afb1a158f3a8701e038be9dd11918b7a33f95d15a0304546a9d1c278b43db615","family_id":"qf-factorial-successor-recurrence","seed":20261041,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":14,\"n_factorial\":\"87178291200\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 14!=87178291200, use (n+1)!=(n+1)×n! to find 15!.","answer":{"kind":"short_text","value":"1307674368000","explanation":"15×87178291200=1307674368000, so 15!=1307674368000."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0016","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96030610578fbf076889c58f54bbc357075b3ef47a4b17b80bb8ff7985712421","family_id":"qf-factorial-successor-recurrence","seed":20261042,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":15,\"n_factorial\":\"1307674368000\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 15!=1307674368000, use (n+1)!=(n+1)×n! to find 16!.","answer":{"kind":"short_text","value":"20922789888000","explanation":"16×1307674368000=20922789888000, so 16!=20922789888000."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0017","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0f9e3fd500852713e085fafabbdacc263146c601fe1c677b4331dded283c7ea","family_id":"qf-factorial-successor-recurrence","seed":20261043,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":16,\"n_factorial\":\"20922789888000\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 16!=20922789888000, use (n+1)!=(n+1)×n! to find 17!.","answer":{"kind":"short_text","value":"355687428096000","explanation":"17×20922789888000=355687428096000, so 17!=355687428096000."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0018","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1abe8cab4a8ae6b63028c6c6ab88d87f215f53fe02fccbeb235a69d10d539997","family_id":"qf-factorial-successor-recurrence","seed":20261044,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":17,\"n_factorial\":\"355687428096000\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 17!=355687428096000, use (n+1)!=(n+1)×n! to find 18!.","answer":{"kind":"short_text","value":"6402373705728000","explanation":"18×355687428096000=6402373705728000, so 18!=6402373705728000."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0019","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be421bdebf786499aaa24a8d5ada6d725c59381564b130937cffda2bf1204432","family_id":"qf-factorial-successor-recurrence","seed":20261045,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":18,\"n_factorial\":\"6402373705728000\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 18!=6402373705728000, use (n+1)!=(n+1)×n! to find 19!.","answer":{"kind":"short_text","value":"121645100408832000","explanation":"19×6402373705728000=121645100408832000, so 19!=121645100408832000."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0020","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c0f9e27b41b8a629a93a37252723711ae0e31320773ee5ce059e2dea1a776cb","family_id":"qf-factorial-successor-recurrence","seed":20261046,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":19,\"n_factorial\":\"121645100408832000\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 19!=121645100408832000, use (n+1)!=(n+1)×n! to find 20!.","answer":{"kind":"short_text","value":"2432902008176640000","explanation":"20×121645100408832000=2432902008176640000, so 20!=2432902008176640000."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0021","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c7d3c78bcf0d3dc17fa112e791468ac5d64035c102beb0e6aadfac7e87930a2","family_id":"qf-factorial-successor-recurrence","seed":20261047,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":20,\"n_factorial\":\"2432902008176640000\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 20!=2432902008176640000, use (n+1)!=(n+1)×n! to find 21!.","answer":{"kind":"short_text","value":"51090942171709440000","explanation":"21×2432902008176640000=51090942171709440000, so 21!=51090942171709440000."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0022","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84db7ba79a6316a430d921e44941984ab95a81b003ffc94db910ec430a860ca8","family_id":"qf-factorial-successor-recurrence","seed":20261048,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":21,\"n_factorial\":\"51090942171709440000\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 21!=51090942171709440000, use (n+1)!=(n+1)×n! to find 22!.","answer":{"kind":"short_text","value":"1124000727777607680000","explanation":"22×51090942171709440000=1124000727777607680000, so 22!=1124000727777607680000."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0023","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d796f860928cac41c6f461fcef55df9c89e874c68d1a661fd06d202a12878366","family_id":"qf-factorial-successor-recurrence","seed":20261049,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":22,\"n_factorial\":\"1124000727777607680000\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 22!=1124000727777607680000, use (n+1)!=(n+1)×n! to find 23!.","answer":{"kind":"short_text","value":"25852016738884976640000","explanation":"23×1124000727777607680000=25852016738884976640000, so 23!=25852016738884976640000."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0024","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30b59e78141b0e7b366a1bf6872a1f63d0d65308ee3d27423b096b47f337c3ad","family_id":"qf-factorial-successor-recurrence","seed":20261050,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":23,\"n_factorial\":\"25852016738884976640000\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 23!=25852016738884976640000, use (n+1)!=(n+1)×n! to find 24!.","answer":{"kind":"short_text","value":"620448401733239439360000","explanation":"24×25852016738884976640000=620448401733239439360000, so 24!=620448401733239439360000."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-successor-recurrence-0025","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21058a429cbc2e253e156ad6feeb9df0e2ee80c8d0e2333a65ee71d6cd07cdf5","family_id":"qf-factorial-successor-recurrence","seed":20261051,"semantic_template":"qf-factorial-successor-recurrence/{\"n\":24,\"n_factorial\":\"620448401733239439360000\",\"operation\":\"successor_recurrence\"}"},"fields":{"title":"Successor Recurrence for Factorial","stem":"Given 24!=620448401733239439360000, use (n+1)!=(n+1)×n! to find 25!.","answer":{"kind":"short_text","value":"15511210043330985984000000","explanation":"25×620448401733239439360000=15511210043330985984000000, so 25!=15511210043330985984000000."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0001","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"242e334236e5d05dc2676c96c5bbf2114e88ab07588444e7cd13a3624b927f96","family_id":"qf-factorial-consecutive-quotient","seed":20261127,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":1,\"n_factorial\":\"1\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"1\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 1!/0!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"1","explanation":"1/1=1, using 1!=1×0!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0002","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9cbaa05b39d75eced8f785bb441181fa69cedddecbb3f32525c53c29fea33972","family_id":"qf-factorial-consecutive-quotient","seed":20261128,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":2,\"n_factorial\":\"2\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"1\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 2!/1!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"2","explanation":"2/1=2, using 2!=2×1!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0003","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96b9a1781db386d0ae5854be8ff7698d042941c03497962cc7e6b9a32f295ff0","family_id":"qf-factorial-consecutive-quotient","seed":20261129,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":3,\"n_factorial\":\"6\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"2\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 3!/2!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"3","explanation":"6/2=3, using 3!=3×2!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0004","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"325a206cf48a9bd0ac712860339c3c912595ba3c4070c011bd4bb4c956fe7b9e","family_id":"qf-factorial-consecutive-quotient","seed":20261130,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":4,\"n_factorial\":\"24\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"6\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 4!/3!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"4","explanation":"24/6=4, using 4!=4×3!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0005","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67672e9701a96cf852342546dde0bbf77d7253b96baff27bdaa092184cc21932","family_id":"qf-factorial-consecutive-quotient","seed":20261131,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":5,\"n_factorial\":\"120\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"24\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 5!/4!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"5","explanation":"120/24=5, using 5!=5×4!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0006","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1cb0e7e8ca5dfc45a18ef558cdfaf6dded6ecd877e2f114fece61e432e0751d","family_id":"qf-factorial-consecutive-quotient","seed":20261132,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":6,\"n_factorial\":\"720\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"120\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 6!/5!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"6","explanation":"720/120=6, using 6!=6×5!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0007","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c6b6b1e13dd397f562e1ab5a88f9ce53558c3d018cf716c2dc21c4e0792ffda","family_id":"qf-factorial-consecutive-quotient","seed":20261133,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":7,\"n_factorial\":\"5040\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"720\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 7!/6!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"7","explanation":"5040/720=7, using 7!=7×6!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0008","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"200a2a645b2090e1e530e57b1bd3b113deb8ab3d8e6ca18f37861858d56f9a7b","family_id":"qf-factorial-consecutive-quotient","seed":20261134,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":8,\"n_factorial\":\"40320\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"5040\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 8!/7!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"8","explanation":"40320/5040=8, using 8!=8×7!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0009","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1487c560463e574eb4d6231124ba58166c0203bb084e8acb371c9077dbfaebe8","family_id":"qf-factorial-consecutive-quotient","seed":20261135,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":9,\"n_factorial\":\"362880\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"40320\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 9!/8!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"9","explanation":"362880/40320=9, using 9!=9×8!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0010","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5f26c52bc100ebb7a0acd06abaa8dfa8c24fd7102ebd7b0d810d6ac77cca614","family_id":"qf-factorial-consecutive-quotient","seed":20261136,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":10,\"n_factorial\":\"3628800\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"362880\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 10!/9!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"10","explanation":"3628800/362880=10, using 10!=10×9!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0011","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"213c96aac24dfae002d25c02dc5789522e183444b69e8b5969b318ec0156999e","family_id":"qf-factorial-consecutive-quotient","seed":20261137,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":11,\"n_factorial\":\"39916800\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"3628800\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 11!/10!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"11","explanation":"39916800/3628800=11, using 11!=11×10!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0012","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2af51cba707d5004e107bf424249cadaadaf23e83e9ae169a8100d294668d947","family_id":"qf-factorial-consecutive-quotient","seed":20261138,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":12,\"n_factorial\":\"479001600\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"39916800\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 12!/11!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"12","explanation":"479001600/39916800=12, using 12!=12×11!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0013","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"171e4d0c550492b2bb518ea9852c24e89ad377f974efc7410717d953638cbc42","family_id":"qf-factorial-consecutive-quotient","seed":20261139,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":13,\"n_factorial\":\"6227020800\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"479001600\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 13!/12!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"13","explanation":"6227020800/479001600=13, using 13!=13×12!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0014","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f31fd802152f68c4951283b0efcda36058c04e66e201cc2c2a2dc67c5357b2f","family_id":"qf-factorial-consecutive-quotient","seed":20261140,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":14,\"n_factorial\":\"87178291200\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"6227020800\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 14!/13!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"14","explanation":"87178291200/6227020800=14, using 14!=14×13!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0015","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"394b5d73ba7f540fd13e925b756e2d5dfa6d5541dc1bcecefbe8a944f4ef0fcd","family_id":"qf-factorial-consecutive-quotient","seed":20261141,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":15,\"n_factorial\":\"1307674368000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"87178291200\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 15!/14!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"15","explanation":"1307674368000/87178291200=15, using 15!=15×14!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0016","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20ed37e19d2b9a99fdcc5a322fdd094fc5f13938d0c8111a0e36000af4be9308","family_id":"qf-factorial-consecutive-quotient","seed":20261142,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":16,\"n_factorial\":\"20922789888000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"1307674368000\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 16!/15!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"16","explanation":"20922789888000/1307674368000=16, using 16!=16×15!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0017","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61a4c3d05b336db658c8ea5d22e0d373e7f33fcad742c567561536306252a392","family_id":"qf-factorial-consecutive-quotient","seed":20261143,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":17,\"n_factorial\":\"355687428096000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"20922789888000\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 17!/16!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"17","explanation":"355687428096000/20922789888000=17, using 17!=17×16!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0018","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3111e5afd80793eee822bc5fae34345660b52f3d3bd8a74b32d74bed69551468","family_id":"qf-factorial-consecutive-quotient","seed":20261144,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":18,\"n_factorial\":\"6402373705728000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"355687428096000\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 18!/17!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"18","explanation":"6402373705728000/355687428096000=18, using 18!=18×17!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0019","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4788b0e8148eb935addd8caf7fb182806ae31ca8d4497ff630fa216d8d52a77","family_id":"qf-factorial-consecutive-quotient","seed":20261145,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":19,\"n_factorial\":\"121645100408832000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"6402373705728000\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 19!/18!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"19","explanation":"121645100408832000/6402373705728000=19, using 19!=19×18!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0020","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc62f82e7a11a6bc5e8e4fee804523d6bdda5a0fecc9f99949e1c5926639b7e8","family_id":"qf-factorial-consecutive-quotient","seed":20261146,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":20,\"n_factorial\":\"2432902008176640000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"121645100408832000\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 20!/19!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"20","explanation":"2432902008176640000/121645100408832000=20, using 20!=20×19!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0021","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12ebfb1c98e1f5f12dcd9e3d113b0cdbca4a683239c1e6201aae7e8f8a82df61","family_id":"qf-factorial-consecutive-quotient","seed":20261147,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":21,\"n_factorial\":\"51090942171709440000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"2432902008176640000\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 21!/20!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"21","explanation":"51090942171709440000/2432902008176640000=21, using 21!=21×20!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0022","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c40d71736375c1a70b7e124f68d34e0a9175b4634c9e8fd5a8b65563d6ea41d","family_id":"qf-factorial-consecutive-quotient","seed":20261148,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":22,\"n_factorial\":\"1124000727777607680000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"51090942171709440000\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 22!/21!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"22","explanation":"1124000727777607680000/51090942171709440000=22, using 22!=22×21!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0023","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb0d7eea58adc2d17ba0143cf0b079e9c92c092641bbd50195f2286579da6520","family_id":"qf-factorial-consecutive-quotient","seed":20261149,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":23,\"n_factorial\":\"25852016738884976640000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"1124000727777607680000\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 23!/22!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"23","explanation":"25852016738884976640000/1124000727777607680000=23, using 23!=23×22!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0024","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8fe1a859c9724a9b63f2a0c04fcbdebfc711d32871eb3c8b331fda6199bbf14","family_id":"qf-factorial-consecutive-quotient","seed":20261150,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":24,\"n_factorial\":\"620448401733239439360000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"25852016738884976640000\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 24!/23!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"24","explanation":"620448401733239439360000/25852016738884976640000=24, using 24!=24×23!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-consecutive-quotient-0025","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1004dfbdf2904010c1a1854ce8ab286168eb50bb7df17ee1eef5e9a97ede371e","family_id":"qf-factorial-consecutive-quotient","seed":20261151,"semantic_template":"qf-factorial-consecutive-quotient/{\"n\":25,\"n_factorial\":\"15511210043330985984000000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"620448401733239439360000\"}"},"fields":{"title":"Quotient of Consecutive Factorials","stem":"Compute 25!/24!; the denominator is explicitly positive.","answer":{"kind":"short_text","value":"25","explanation":"15511210043330985984000000/620448401733239439360000=25, using 25!=25×24!."},"skills":["factorial quotient","positive divisor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0001","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5844311e34c43468569d37f68c49f271550a77bdac18262591916866ec869b51","family_id":"qf-factorial-missing-successor-factor","seed":20261227,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":0,\"n_factorial\":\"1\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"1\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 1!=□×0!, where 0!=1.","answer":{"kind":"short_text","value":"1","explanation":"1= 1×1, so □=1."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0002","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4af78eb748a3c0563f0711d56838d4c4ec75a001d2d0018974d43935044dd08b","family_id":"qf-factorial-missing-successor-factor","seed":20261228,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":1,\"n_factorial\":\"1\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"2\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 2!=□×1!, where 1!=1.","answer":{"kind":"short_text","value":"2","explanation":"2= 2×1, so □=2."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0003","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b596d9f1aaa1966fddbe031ec261a4a1d01c8c015f808f7844110bf79d710b2","family_id":"qf-factorial-missing-successor-factor","seed":20261229,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":2,\"n_factorial\":\"2\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"6\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 3!=□×2!, where 2!=2.","answer":{"kind":"short_text","value":"3","explanation":"6= 3×2, so □=3."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0004","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61b41dbad4c69377a46591d932509b06cbb2f0a3e9da1c888430d99a95fa0abc","family_id":"qf-factorial-missing-successor-factor","seed":20261230,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":3,\"n_factorial\":\"6\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"24\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 4!=□×3!, where 3!=6.","answer":{"kind":"short_text","value":"4","explanation":"24= 4×6, so □=4."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0005","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7976c44cc19f6fa5593772046adacfced4ce946a4ff12af34fd4f53f8f109b9","family_id":"qf-factorial-missing-successor-factor","seed":20261231,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":4,\"n_factorial\":\"24\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"120\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 5!=□×4!, where 4!=24.","answer":{"kind":"short_text","value":"5","explanation":"120= 5×24, so □=5."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0006","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2580b1fd0bbea877085daf2c846beced6339a6b9dd989aaf686ab4c00d747822","family_id":"qf-factorial-missing-successor-factor","seed":20261232,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":5,\"n_factorial\":\"120\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"720\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 6!=□×5!, where 5!=120.","answer":{"kind":"short_text","value":"6","explanation":"720= 6×120, so □=6."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0007","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb3b54da959a9d38b0c9cbaedb5f97c099138f85cb39ca76c02f50067be63b9f","family_id":"qf-factorial-missing-successor-factor","seed":20261233,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":6,\"n_factorial\":\"720\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"5040\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 7!=□×6!, where 6!=720.","answer":{"kind":"short_text","value":"7","explanation":"5040= 7×720, so □=7."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0008","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eddfef550102f6539ed39c6b972e1785dbc3c8472599367c14d6da60018b168e","family_id":"qf-factorial-missing-successor-factor","seed":20261234,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":7,\"n_factorial\":\"5040\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"40320\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 8!=□×7!, where 7!=5040.","answer":{"kind":"short_text","value":"8","explanation":"40320= 8×5040, so □=8."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0009","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7b94a3f91fed5b83e25ec475d7af8658a8bbfc84675c53fac623efd5c632175","family_id":"qf-factorial-missing-successor-factor","seed":20261235,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":8,\"n_factorial\":\"40320\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"362880\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 9!=□×8!, where 8!=40320.","answer":{"kind":"short_text","value":"9","explanation":"362880= 9×40320, so □=9."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0010","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b03c43ae3068a41421c093ad879e012ad5ba5e65df313bf562c7cf238c1df6d","family_id":"qf-factorial-missing-successor-factor","seed":20261236,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":9,\"n_factorial\":\"362880\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"3628800\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 10!=□×9!, where 9!=362880.","answer":{"kind":"short_text","value":"10","explanation":"3628800= 10×362880, so □=10."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0011","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9238b388e8e8b09ecce3518824ef5d3b4f457bb8341e3405eaf2ad1275189335","family_id":"qf-factorial-missing-successor-factor","seed":20261237,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":10,\"n_factorial\":\"3628800\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"39916800\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 11!=□×10!, where 10!=3628800.","answer":{"kind":"short_text","value":"11","explanation":"39916800= 11×3628800, so □=11."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0012","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"680f252f17f2df3c27931aa0eb7f7e3618e6cb0b5b9dfa906b25ab50d53a0e95","family_id":"qf-factorial-missing-successor-factor","seed":20261238,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":11,\"n_factorial\":\"39916800\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"479001600\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 12!=□×11!, where 11!=39916800.","answer":{"kind":"short_text","value":"12","explanation":"479001600= 12×39916800, so □=12."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0013","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5db53864f128ed3ec86d3dc28a17219f43a7f3b37dc6247cc7120bade0f1f450","family_id":"qf-factorial-missing-successor-factor","seed":20261239,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":12,\"n_factorial\":\"479001600\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"6227020800\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 13!=□×12!, where 12!=479001600.","answer":{"kind":"short_text","value":"13","explanation":"6227020800= 13×479001600, so □=13."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0014","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a26666f863474a35499aa918cc7473904f6fd5e96d6cdad8863d4bf8779b26e8","family_id":"qf-factorial-missing-successor-factor","seed":20261240,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":13,\"n_factorial\":\"6227020800\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"87178291200\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 14!=□×13!, where 13!=6227020800.","answer":{"kind":"short_text","value":"14","explanation":"87178291200= 14×6227020800, so □=14."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0015","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a11ac540ee591f4b982dade931fefea451796fb02c5f2cf12ef384fcde03d1e","family_id":"qf-factorial-missing-successor-factor","seed":20261241,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":14,\"n_factorial\":\"87178291200\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"1307674368000\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 15!=□×14!, where 14!=87178291200.","answer":{"kind":"short_text","value":"15","explanation":"1307674368000= 15×87178291200, so □=15."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0016","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49d108222cd981d4d5f890c6284af3287495fc4e25117107bbd25b46c244e05c","family_id":"qf-factorial-missing-successor-factor","seed":20261242,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":15,\"n_factorial\":\"1307674368000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"20922789888000\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 16!=□×15!, where 15!=1307674368000.","answer":{"kind":"short_text","value":"16","explanation":"20922789888000= 16×1307674368000, so □=16."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0017","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02e3fc192d7224c5e40746ad98798fa9e25093045d5e42804f27ddadbc304b14","family_id":"qf-factorial-missing-successor-factor","seed":20261243,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":16,\"n_factorial\":\"20922789888000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"355687428096000\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 17!=□×16!, where 16!=20922789888000.","answer":{"kind":"short_text","value":"17","explanation":"355687428096000= 17×20922789888000, so □=17."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0018","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6d0d3be5aaccb7dd1b2726b97ed3a54b3724700e432ad6c70303845e826a83d","family_id":"qf-factorial-missing-successor-factor","seed":20261244,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":17,\"n_factorial\":\"355687428096000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"6402373705728000\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 18!=□×17!, where 17!=355687428096000.","answer":{"kind":"short_text","value":"18","explanation":"6402373705728000= 18×355687428096000, so □=18."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0019","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01d74fb8bdbf90136b58f62f1013f58ef66e17ebb5a688ad5f11c5cf38fc6b15","family_id":"qf-factorial-missing-successor-factor","seed":20261245,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":18,\"n_factorial\":\"6402373705728000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"121645100408832000\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 19!=□×18!, where 18!=6402373705728000.","answer":{"kind":"short_text","value":"19","explanation":"121645100408832000= 19×6402373705728000, so □=19."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0020","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42b5cd3e4d286a10d34e084eb638ab538642e99ea0612078ec733e5d009be162","family_id":"qf-factorial-missing-successor-factor","seed":20261246,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":19,\"n_factorial\":\"121645100408832000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"2432902008176640000\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 20!=□×19!, where 19!=121645100408832000.","answer":{"kind":"short_text","value":"20","explanation":"2432902008176640000= 20×121645100408832000, so □=20."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0021","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b168ada652bb6a059e48cadcfae4d2af068714793aafe7d78df5c1a8ed39a3eb","family_id":"qf-factorial-missing-successor-factor","seed":20261247,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":20,\"n_factorial\":\"2432902008176640000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"51090942171709440000\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 21!=□×20!, where 20!=2432902008176640000.","answer":{"kind":"short_text","value":"21","explanation":"51090942171709440000= 21×2432902008176640000, so □=21."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0022","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"846ce7a19e34175fd502055acd0df462484c9001e4873ce2977dea1c460c175b","family_id":"qf-factorial-missing-successor-factor","seed":20261248,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":21,\"n_factorial\":\"51090942171709440000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"1124000727777607680000\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 22!=□×21!, where 21!=51090942171709440000.","answer":{"kind":"short_text","value":"22","explanation":"1124000727777607680000= 22×51090942171709440000, so □=22."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0023","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fdaf0af23b7ce37f55f7061a91a10848a8eb48bedba6eec3346d1860db01670","family_id":"qf-factorial-missing-successor-factor","seed":20261249,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":22,\"n_factorial\":\"1124000727777607680000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"25852016738884976640000\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 23!=□×22!, where 22!=1124000727777607680000.","answer":{"kind":"short_text","value":"23","explanation":"25852016738884976640000= 23×1124000727777607680000, so □=23."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0024","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9d2aecc8e92f29b7ba40401548af5d573cc93e749bb5a5b62b9fe70fd8f72fe","family_id":"qf-factorial-missing-successor-factor","seed":20261250,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":23,\"n_factorial\":\"25852016738884976640000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"620448401733239439360000\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 24!=□×23!, where 23!=25852016738884976640000.","answer":{"kind":"short_text","value":"24","explanation":"620448401733239439360000= 24×25852016738884976640000, so □=24."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260926-factorial-missing-successor-factor-0025","batch_id":"batch-20260926-factorial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d82bc874d3cf49145c9f75e2b30e65ca5564adb6dc2f938362f5541eff5d7909","family_id":"qf-factorial-missing-successor-factor","seed":20261251,"semantic_template":"qf-factorial-missing-successor-factor/{\"n\":24,\"n_factorial\":\"620448401733239439360000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"15511210043330985984000000\"}"},"fields":{"title":"Missing Factor in the Successor Recurrence","stem":"Fill the integer □ in 25!=□×24!, where 24!=620448401733239439360000.","answer":{"kind":"short_text","value":"25","explanation":"15511210043330985984000000= 25×620448401733239439360000, so □=25."},"skills":["factorial recurrence","BigInt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}