第 27461 題：整數p進賦值

原題與已知條件

prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={"p":11,"a":121,"b":-1452}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={"p":11,"a":121,"b":-1452}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

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-p-adic-valuation-integers

逐步推導

source roles = [pval-product-additivity]

derived = [[p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention]]

Valuation ORIGINAL a ORIGINAL division step1 := div(121, 11)

Valuation ORIGINAL a ORIGINAL division step1 = 11

Valuation ORIGINAL a dividing prime power1 := mul(1, 11)

Valuation ORIGINAL a dividing prime power1 = 11

source roles = [Valuation ORIGINAL a, 11, 121, 121, 1]

derived = [11, 11, 0, 是]

Valuation ORIGINAL a ORIGINAL division step2 := div(11, 11)

Valuation ORIGINAL a ORIGINAL division step2 = 1

Valuation ORIGINAL a dividing prime power2 := mul(11, 11)

Valuation ORIGINAL a dividing prime power2 = 121

source roles = [Valuation ORIGINAL a, 11, 121, 11, 2]

derived = [1, 121, 0, 是]

Valuation ORIGINAL a terminal factorization product := mul(121, 1)

Valuation ORIGINAL a terminal factorization product = 121

source roles = [Valuation ORIGINAL a, 11, 121, 121, 2, 1, [11, 121]]

derived = [121, 1, 是, 是]

Valuation ORIGINAL b ORIGINAL division step1 := div(1452, 11)

Valuation ORIGINAL b ORIGINAL division step1 = 132

Valuation ORIGINAL b dividing prime power1 := mul(1, 11)

Valuation ORIGINAL b dividing prime power1 = 11

source roles = [Valuation ORIGINAL b, 11, 1452, 1452, 1]

derived = [132, 11, 0, 是]

Valuation ORIGINAL b ORIGINAL division step2 := div(132, 11)

Valuation ORIGINAL b ORIGINAL division step2 = 12

Valuation ORIGINAL b dividing prime power2 := mul(11, 11)

Valuation ORIGINAL b dividing prime power2 = 121

source roles = [Valuation ORIGINAL b, 11, 1452, 132, 2]

derived = [12, 121, 0, 是]

Valuation ORIGINAL b terminal factorization product := mul(121, 12)

Valuation ORIGINAL b terminal factorization product = 1452

source roles = [Valuation ORIGINAL b, 11, -1452, 1452, 2, 12, [11, 121]]

derived = [1452, 1, 是, 是]

Valuation ORIGINAL a times b := mul(121, -1452)

Valuation ORIGINAL a times b = -175692

Valuation ACTUAL product ORIGINAL division step1 := div(175692, 11)

Valuation ACTUAL product ORIGINAL division step1 = 15972

Valuation ACTUAL product dividing prime power1 := mul(1, 11)

Valuation ACTUAL product dividing prime power1 = 11

source roles = [Valuation ACTUAL product, 11, 175692, 175692, 1]

derived = [15972, 11, 0, 是]

Valuation ACTUAL product ORIGINAL division step2 := div(15972, 11)

Valuation ACTUAL product ORIGINAL division step2 = 1452

Valuation ACTUAL product dividing prime power2 := mul(11, 11)

Valuation ACTUAL product dividing prime power2 = 121

source roles = [Valuation ACTUAL product, 11, 175692, 15972, 2]

derived = [1452, 121, 0, 是]

Valuation ACTUAL product ORIGINAL division step3 := div(1452, 11)

Valuation ACTUAL product ORIGINAL division step3 = 132

Valuation ACTUAL product dividing prime power3 := mul(121, 11)

Valuation ACTUAL product dividing prime power3 = 1331

source roles = [Valuation ACTUAL product, 11, 175692, 1452, 3]

derived = [132, 1331, 0, 是]

Valuation ACTUAL product ORIGINAL division step4 := div(132, 11)

Valuation ACTUAL product ORIGINAL division step4 = 12

Valuation ACTUAL product dividing prime power4 := mul(1331, 11)

Valuation ACTUAL product dividing prime power4 = 14641

source roles = [Valuation ACTUAL product, 11, 175692, 132, 4]

derived = [12, 14641, 0, 是]

Valuation ACTUAL product terminal factorization product := mul(14641, 12)

Valuation ACTUAL product terminal factorization product = 175692

source roles = [Valuation ACTUAL product, 11, -175692, 175692, 4, 12, [11, 121, 1331, 14641]]

derived = [175692, 1, 是, 是]

Valuation ORIGINAL exponent addition := sum(2, 2)

Valuation ORIGINAL exponent addition = 4

source roles = [11, 121, -1452, 2, 2, 1, 12]

derived = [-175692, 4, 4, 是]

逐步反向核對

• S034 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S033 · premise_and_result_audit — 以精確有理數重算 Valuation ORIGINAL exponent addition，不以小數近似替代。
• S032 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S031 · premise_and_result_audit — 以精確有理數重算 Valuation ACTUAL product terminal factorization product，不以小數近似替代。
• S030 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S029 · premise_and_result_audit — 以精確有理數重算 Valuation ACTUAL product dividing prime power4，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Valuation ACTUAL product ORIGINAL division step4，不以小數近似替代。
• S027 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S026 · premise_and_result_audit — 以精確有理數重算 Valuation ACTUAL product dividing prime power3，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Valuation ACTUAL product ORIGINAL division step3，不以小數近似替代。
• S024 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S023 · premise_and_result_audit — 以精確有理數重算 Valuation ACTUAL product dividing prime power2，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Valuation ACTUAL product ORIGINAL division step2，不以小數近似替代。
• S021 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S020 · premise_and_result_audit — 以精確有理數重算 Valuation ACTUAL product dividing prime power1，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Valuation ACTUAL product ORIGINAL division step1，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Valuation ORIGINAL a times b，不以小數近似替代。
• S017 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S016 · premise_and_result_audit — 以精確有理數重算 Valuation ORIGINAL b terminal factorization product，不以小數近似替代。
• S015 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S014 · premise_and_result_audit — 以精確有理數重算 Valuation ORIGINAL b dividing prime power2，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Valuation ORIGINAL b ORIGINAL division step2，不以小數近似替代。
• S012 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S011 · premise_and_result_audit — 以精確有理數重算 Valuation ORIGINAL b dividing prime power1，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Valuation ORIGINAL b ORIGINAL division step1，不以小數近似替代。
• S009 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S008 · premise_and_result_audit — 以精確有理數重算 Valuation ORIGINAL a terminal factorization product，不以小數近似替代。
• S007 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S006 · premise_and_result_audit — 以精確有理數重算 Valuation ORIGINAL a dividing prime power2，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Valuation ORIGINAL a ORIGINAL division step2，不以小數近似替代。
• S004 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S003 · premise_and_result_audit — 以精確有理數重算 Valuation ORIGINAL a dividing prime power1，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Valuation ORIGINAL a ORIGINAL division step1，不以小數近似替代。
• S001 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。

結論

(11, 121, -1452, 2, 2, -175692, 4, 4, 是, infinity_plus_k_equals_infinity)

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。