第 31818 題：連續階乘的商

原題與已知條件

計算 18!/17!；分母已明示為正。

• G001 — 計算 18!/17!；分母已明示為正。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-factorial

逐步推導

source roles = [qf-factorial-consecutive-quotient, no-domain-field, consecutive_quotient, 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

source roles = [Factorial requested n, 18]

derived = [6402373705728000]

source roles = [18, 6402373705728000]

derived = [6402373705728000]

Factorial previous n minus1 ORIGINAL factorial factor2 := mul(1, 2)

Factorial previous n minus1 ORIGINAL factorial factor2 = 2

Factorial previous n minus1 ORIGINAL factorial factor3 := mul(2, 3)

Factorial previous n minus1 ORIGINAL factorial factor3 = 6

Factorial previous n minus1 ORIGINAL factorial factor4 := mul(6, 4)

Factorial previous n minus1 ORIGINAL factorial factor4 = 24

Factorial previous n minus1 ORIGINAL factorial factor5 := mul(24, 5)

Factorial previous n minus1 ORIGINAL factorial factor5 = 120

Factorial previous n minus1 ORIGINAL factorial factor6 := mul(120, 6)

Factorial previous n minus1 ORIGINAL factorial factor6 = 720

Factorial previous n minus1 ORIGINAL factorial factor7 := mul(720, 7)

Factorial previous n minus1 ORIGINAL factorial factor7 = 5040

Factorial previous n minus1 ORIGINAL factorial factor8 := mul(5040, 8)

Factorial previous n minus1 ORIGINAL factorial factor8 = 40320

Factorial previous n minus1 ORIGINAL factorial factor9 := mul(40320, 9)

Factorial previous n minus1 ORIGINAL factorial factor9 = 362880

Factorial previous n minus1 ORIGINAL factorial factor10 := mul(362880, 10)

Factorial previous n minus1 ORIGINAL factorial factor10 = 3628800

Factorial previous n minus1 ORIGINAL factorial factor11 := mul(3628800, 11)

Factorial previous n minus1 ORIGINAL factorial factor11 = 39916800

Factorial previous n minus1 ORIGINAL factorial factor12 := mul(39916800, 12)

Factorial previous n minus1 ORIGINAL factorial factor12 = 479001600

Factorial previous n minus1 ORIGINAL factorial factor13 := mul(479001600, 13)

Factorial previous n minus1 ORIGINAL factorial factor13 = 6227020800

Factorial previous n minus1 ORIGINAL factorial factor14 := mul(6227020800, 14)

Factorial previous n minus1 ORIGINAL factorial factor14 = 87178291200

Factorial previous n minus1 ORIGINAL factorial factor15 := mul(87178291200, 15)

Factorial previous n minus1 ORIGINAL factorial factor15 = 1307674368000

Factorial previous n minus1 ORIGINAL factorial factor16 := mul(1307674368000, 16)

Factorial previous n minus1 ORIGINAL factorial factor16 = 20922789888000

Factorial previous n minus1 ORIGINAL factorial factor17 := mul(20922789888000, 17)

Factorial previous n minus1 ORIGINAL factorial factor17 = 355687428096000

source roles = [Factorial previous n minus1, 17]

derived = [355687428096000]

Factorial ORIGINAL n factorial over positive previous factorial := div(6402373705728000, 355687428096000)

Factorial ORIGINAL n factorial over positive previous factorial = 18

Factorial quotient reverse ORIGINAL previous factorial times n := mul(355687428096000, 18)

Factorial quotient reverse ORIGINAL previous factorial times n = 6402373705728000

source roles = [18, 6402373705728000, 355687428096000]

derived = [18, 6402373705728000]

逐步反向核對

• S040 · premise_and_result_audit — 反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。
• S039 · premise_and_result_audit — 以精確有理數重算 Factorial quotient reverse ORIGINAL previous factorial times n，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Factorial ORIGINAL n factorial over positive previous factorial，不以小數近似替代。
• S037 · premise_and_result_audit — 反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。
• S036 · premise_and_result_audit — 以精確有理數重算 Factorial previous n minus1 ORIGINAL factorial factor17，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Factorial previous n minus1 ORIGINAL factorial factor16，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Factorial previous n minus1 ORIGINAL factorial factor15，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Factorial previous n minus1 ORIGINAL factorial factor14，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Factorial previous n minus1 ORIGINAL factorial factor13，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Factorial previous n minus1 ORIGINAL factorial factor12，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Factorial previous n minus1 ORIGINAL factorial factor11，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Factorial previous n minus1 ORIGINAL factorial factor10，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Factorial previous n minus1 ORIGINAL factorial factor9，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Factorial previous n minus1 ORIGINAL factorial factor8，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Factorial previous n minus1 ORIGINAL factorial factor7，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Factorial previous n minus1 ORIGINAL factorial factor6，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Factorial previous n minus1 ORIGINAL factorial factor5，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Factorial previous n minus1 ORIGINAL factorial factor4，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Factorial previous n minus1 ORIGINAL factorial factor3，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Factorial previous n minus1 ORIGINAL factorial factor2，不以小數近似替代。
• S020 · premise_and_result_audit — 反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。
• S019 · premise_and_result_audit — 反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。
• S018 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor18，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor17，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor16，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor15，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor14，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor13，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor12，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor11，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor10，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor9，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor8，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor7，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor6，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor5，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor4，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor3，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor2，不以小數近似替代。
• S001 · premise_and_result_audit — 反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。

結論

18

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。