Question 31871: Missing Factor in the Successor Recurrence

Original question and givens

Fill the integer □ in 21!=□×20!, where 20!=2432902008176640000.

• G001 — Fill the integer □ in 21!=□×20!, where 20!=2432902008176640000.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-factorial

Step-by-step derivation

source roles = [qf-factorial-missing-successor-factor, no-domain-field, missing_successor_factor, no-form-field]

derived = [[exact_integer_STRING]]

Factorial requested n ORIGINAL factorial factor2 := mul(1, 2)

Factorial requested n ORIGINAL factorial factor2 = 2

Factorial requested n ORIGINAL factorial factor3 := mul(2, 3)

Factorial requested n ORIGINAL factorial factor3 = 6

Factorial requested n ORIGINAL factorial factor4 := mul(6, 4)

Factorial requested n ORIGINAL factorial factor4 = 24

Factorial requested n ORIGINAL factorial factor5 := mul(24, 5)

Factorial requested n ORIGINAL factorial factor5 = 120

Factorial requested n ORIGINAL factorial factor6 := mul(120, 6)

Factorial requested n ORIGINAL factorial factor6 = 720

Factorial requested n ORIGINAL factorial factor7 := mul(720, 7)

Factorial requested n ORIGINAL factorial factor7 = 5040

Factorial requested n ORIGINAL factorial factor8 := mul(5040, 8)

Factorial requested n ORIGINAL factorial factor8 = 40320

Factorial requested n ORIGINAL factorial factor9 := mul(40320, 9)

Factorial requested n ORIGINAL factorial factor9 = 362880

Factorial requested n ORIGINAL factorial factor10 := mul(362880, 10)

Factorial requested n ORIGINAL factorial factor10 = 3628800

Factorial requested n ORIGINAL factorial factor11 := mul(3628800, 11)

Factorial requested n ORIGINAL factorial factor11 = 39916800

Factorial requested n ORIGINAL factorial factor12 := mul(39916800, 12)

Factorial requested n ORIGINAL factorial factor12 = 479001600

Factorial requested n ORIGINAL factorial factor13 := mul(479001600, 13)

Factorial requested n ORIGINAL factorial factor13 = 6227020800

Factorial requested n ORIGINAL factorial factor14 := mul(6227020800, 14)

Factorial requested n ORIGINAL factorial factor14 = 87178291200

Factorial requested n ORIGINAL factorial factor15 := mul(87178291200, 15)

Factorial requested n ORIGINAL factorial factor15 = 1307674368000

Factorial requested n ORIGINAL factorial factor16 := mul(1307674368000, 16)

Factorial requested n ORIGINAL factorial factor16 = 20922789888000

Factorial requested n ORIGINAL factorial factor17 := mul(20922789888000, 17)

Factorial requested n ORIGINAL factorial factor17 = 355687428096000

Factorial requested n ORIGINAL factorial factor18 := mul(355687428096000, 18)

Factorial requested n ORIGINAL factorial factor18 = 6402373705728000

Factorial requested n ORIGINAL factorial factor19 := mul(6402373705728000, 19)

Factorial requested n ORIGINAL factorial factor19 = 121645100408832000

Factorial requested n ORIGINAL factorial factor20 := mul(121645100408832000, 20)

Factorial requested n ORIGINAL factorial factor20 = 2432902008176640000

source roles = [Factorial requested n, 20]

derived = [2432902008176640000]

source roles = [20, 2432902008176640000]

derived = [2432902008176640000]

Factorial ORIGINAL successor index n plus1 := sum(20, 1)

Factorial ORIGINAL successor index n plus1 = 21

Factorial ORIGINAL successor factor times verified n factorial := mul(21, 2432902008176640000)

Factorial ORIGINAL successor factor times verified n factorial = 51090942171709440000

Factorial missing ORIGINAL verified successor over positive n factorial := div(51090942171709440000, 2432902008176640000)

Factorial missing ORIGINAL verified successor over positive n factorial = 21

source roles = [20, 2432902008176640000, 51090942171709440000]

derived = [21]

Reverse checks

• S026 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
• S025 · premise_and_result_audit — Recompute Factorial missing ORIGINAL verified successor over positive n factorial with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Factorial ORIGINAL successor factor times verified n factorial with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Factorial ORIGINAL successor index n plus1 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
• S021 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
• S020 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor20 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor19 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor18 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor17 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor16 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor15 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor14 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor13 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor12 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor11 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor10 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor9 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor8 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor7 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor6 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor5 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor4 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.

Result

21

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.