# Question 27432: The Highest Dividing Prime Power Is Extracted from the Absolute Integer [Return to catalogue page 138](https://ocme.evemisslab.com/en/questions/page/138/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations/#q-20261015-advance-20261118-r05-p-adic-valuation-integers-0007) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0053-27432 · q-20261015-advance-20261118-r05-p-adic-valuation-integers-0007 ## Original question and givens Use prime p=2, nonzero integer a=14. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p). - G001 — Use prime p=2, nonzero integer a=14. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p). ## 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-p-adic-valuation-integers](https://ocme.evemisslab.com/en/mko/mko-p-adic-valuation-integers/) ## Step-by-step derivation S001 · impliesBefore source roles = [pval-factor-extraction] After derived = [[p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p]] Conditions: none List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore Valuation ORIGINAL a ORIGINAL division step1 := div(14, 2) After Valuation ORIGINAL a ORIGINAL division step1 = 7 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Valuation ORIGINAL a dividing prime power1 := mul(1, 2) After Valuation ORIGINAL a dividing prime power1 = 2 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore source roles = [Valuation ORIGINAL a, 2, 14, 14, 1] After derived = [7, 2, 0, Yes] Conditions: none Divide only when the current unit is divisible by p.Each step preserves |a|=p^k*unit and strictly decreases the positive unit,so it terminates.Derived integers do not inherit the raw-input1e6 cap. Step identifiers and dependenciesRule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S003 S005 · calculationBefore Valuation ORIGINAL a terminal factorization product := mul(2, 7) After Valuation ORIGINAL a terminal factorization product = 14 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [Valuation ORIGINAL a, 2, 14, 14, 1, 7, [2]] After derived = [14, 1, Yes, Yes] Conditions: none At termination p does not divide the unit,so p^(k+1) does not divide |a| and k is maximal.A sign does not affect divisibility;the factorization is for |a|,not negative a equalling a positive product. Step identifiers and dependenciesRule: Valuation_terminal_NONdivisible_unit_highest_power · Uses: G001, S005 ## Reverse checks - S006 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct. - S005 · premise_and_result_audit — Recompute Valuation ORIGINAL a terminal factorization product with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct. - S003 · premise_and_result_audit — Recompute Valuation ORIGINAL a dividing prime power1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Valuation ORIGINAL a ORIGINAL division step1 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct. ## Result (2, 14, 14, 1, 7, [2], Yes, Yes) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 6658b54e2c9090c550e96ff166deac32dfb52619b7fb6b181d6ee9cf857358a7 - candidate record SHA-256: f68336c8dc289ba4b3550115a55dd9da4ebf0fbd6e02dc6749253169e8c5b2c7 - semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62 - question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0007.json)