# Question 27458: Integer Valuations Add under Products, Including the Fixed Zero Convention [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-0033) 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. R0053-27458 · q-20261015-advance-20261118-r05-p-adic-valuation-integers-0033 ## Original question and givens Use prime p=3, integers a=24, b=-81, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention). - G001 — Use prime p=3, integers a=24, b=-81, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention). ## 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-product-additivity] After derived = [[p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention]] 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(24, 3) After Valuation ORIGINAL a ORIGINAL division step1 = 8 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, 3) After Valuation ORIGINAL a dividing prime power1 = 3 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, 3, 24, 24, 1] After derived = [8, 3, 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(3, 8) After Valuation ORIGINAL a terminal factorization product = 24 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, 3, 24, 24, 1, 8, [3]] After derived = [24, 2, 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 S007 · calculationBefore Valuation ORIGINAL b ORIGINAL division step1 := div(81, 3) After Valuation ORIGINAL b ORIGINAL division step1 = 27 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, S006 S008 · calculationBefore Valuation ORIGINAL b dividing prime power1 := mul(1, 3) After Valuation ORIGINAL b dividing prime power1 = 3 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, S007 S009 · impliesBefore source roles = [Valuation ORIGINAL b, 3, 81, 81, 1] After derived = [27, 3, 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, S008 S010 · calculationBefore Valuation ORIGINAL b ORIGINAL division step2 := div(27, 3) After Valuation ORIGINAL b ORIGINAL division step2 = 9 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, S009 S011 · calculationBefore Valuation ORIGINAL b dividing prime power2 := mul(3, 3) After Valuation ORIGINAL b dividing prime power2 = 9 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, S010 S012 · impliesBefore source roles = [Valuation ORIGINAL b, 3, 81, 27, 2] After derived = [9, 9, 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, S011 S013 · calculationBefore Valuation ORIGINAL b ORIGINAL division step3 := div(9, 3) After Valuation ORIGINAL b ORIGINAL division step3 = 3 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, S012 S014 · calculationBefore Valuation ORIGINAL b dividing prime power3 := mul(9, 3) After Valuation ORIGINAL b dividing prime power3 = 27 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, S013 S015 · impliesBefore source roles = [Valuation ORIGINAL b, 3, 81, 9, 3] After derived = [3, 27, 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, S014 S016 · calculationBefore Valuation ORIGINAL b ORIGINAL division step4 := div(3, 3) After Valuation ORIGINAL b ORIGINAL division step4 = 1 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, S015 S017 · calculationBefore Valuation ORIGINAL b dividing prime power4 := mul(27, 3) After Valuation ORIGINAL b dividing prime power4 = 81 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, S016 S018 · impliesBefore source roles = [Valuation ORIGINAL b, 3, 81, 3, 4] After derived = [1, 81, 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, S017 S019 · calculationBefore Valuation ORIGINAL b terminal factorization product := mul(81, 1) After Valuation ORIGINAL b terminal factorization product = 81 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, S018 S020 · impliesBefore source roles = [Valuation ORIGINAL b, 3, -81, 81, 4, 1, [3, 9, 27, 81]] After derived = [81, 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, S019 S021 · calculationBefore Valuation ORIGINAL a times b := mul(24, -81) After Valuation ORIGINAL a times b = -1944 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, S020 S022 · calculationBefore Valuation ACTUAL product ORIGINAL division step1 := div(1944, 3) After Valuation ACTUAL product ORIGINAL division step1 = 648 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, S021 S023 · calculationBefore Valuation ACTUAL product dividing prime power1 := mul(1, 3) After Valuation ACTUAL product dividing prime power1 = 3 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, S022 S024 · impliesBefore source roles = [Valuation ACTUAL product, 3, 1944, 1944, 1] After derived = [648, 3, 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, S023 S025 · calculationBefore Valuation ACTUAL product ORIGINAL division step2 := div(648, 3) After Valuation ACTUAL product ORIGINAL division step2 = 216 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, S024 S026 · calculationBefore Valuation ACTUAL product dividing prime power2 := mul(3, 3) After Valuation ACTUAL product dividing prime power2 = 9 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, S025 S027 · impliesBefore source roles = [Valuation ACTUAL product, 3, 1944, 648, 2] After derived = [216, 9, 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, S026 S028 · calculationBefore Valuation ACTUAL product ORIGINAL division step3 := div(216, 3) After Valuation ACTUAL product ORIGINAL division step3 = 72 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, S027 S029 · calculationBefore Valuation ACTUAL product dividing prime power3 := mul(9, 3) After Valuation ACTUAL product dividing prime power3 = 27 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, S028 S030 · impliesBefore source roles = [Valuation ACTUAL product, 3, 1944, 216, 3] After derived = [72, 27, 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, S029 S031 · calculationBefore Valuation ACTUAL product ORIGINAL division step4 := div(72, 3) After Valuation ACTUAL product ORIGINAL division step4 = 24 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, S030 S032 · calculationBefore Valuation ACTUAL product dividing prime power4 := mul(27, 3) After Valuation ACTUAL product dividing prime power4 = 81 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, S031 S033 · impliesBefore source roles = [Valuation ACTUAL product, 3, 1944, 72, 4] After derived = [24, 81, 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, S032 S034 · calculationBefore Valuation ACTUAL product ORIGINAL division step5 := div(24, 3) After Valuation ACTUAL product ORIGINAL division step5 = 8 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, S033 S035 · calculationBefore Valuation ACTUAL product dividing prime power5 := mul(81, 3) After Valuation ACTUAL product dividing prime power5 = 243 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, S034 S036 · impliesBefore source roles = [Valuation ACTUAL product, 3, 1944, 24, 5] After derived = [8, 243, 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, S035 S037 · calculationBefore Valuation ACTUAL product terminal factorization product := mul(243, 8) After Valuation ACTUAL product terminal factorization product = 1944 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, S036 S038 · impliesBefore source roles = [Valuation ACTUAL product, 3, -1944, 1944, 5, 8, [3, 9, 27, 81, 243]] After derived = [1944, 2, 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, S037 S039 · calculationBefore Valuation ORIGINAL exponent addition := sum(1, 4) After Valuation ORIGINAL exponent addition = 5 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, S038 S040 · impliesBefore source roles = [3, 24, -81, 1, 4, 8, 1] After derived = [-1944, 5, 5, Yes] Conditions: none For nonzero factors write a=±p^r*u,b=±p^s*v.Primality and Euclid lemma keep uv indivisible by p,so the exact exponent is r+s.If either factor is0,the product is0 and the same infinity convention applies.A derived product may reach1e12 while remaining SAFE;do not reapply the raw1e6 bound. Step identifiers and dependenciesRule: Valuation_GENERAL_product_primeness_unit_product_or_zero · Uses: G001, S039 ## Reverse checks - S040 · 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. - S039 · premise_and_result_audit — Recompute Valuation ORIGINAL exponent addition with exact rational arithmetic, not rounded decimals. - S038 · 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. - S037 · premise_and_result_audit — Recompute Valuation ACTUAL product terminal factorization product with exact rational arithmetic, not rounded decimals. - S036 · 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. - S035 · premise_and_result_audit — Recompute Valuation ACTUAL product dividing prime power5 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute Valuation ACTUAL product ORIGINAL division step5 with exact rational arithmetic, not rounded decimals. - S033 · 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. - S032 · premise_and_result_audit — Recompute Valuation ACTUAL product dividing prime power4 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Valuation ACTUAL product ORIGINAL division step4 with exact rational arithmetic, not rounded decimals. - S030 · 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. - S029 · premise_and_result_audit — Recompute Valuation ACTUAL product dividing prime power3 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute Valuation ACTUAL product ORIGINAL division step3 with exact rational arithmetic, not rounded decimals. - S027 · 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. - S026 · premise_and_result_audit — Recompute Valuation ACTUAL product dividing prime power2 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Valuation ACTUAL product ORIGINAL division step2 with exact rational arithmetic, not rounded decimals. - S024 · 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. - S023 · premise_and_result_audit — Recompute Valuation ACTUAL product dividing prime power1 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Valuation ACTUAL product ORIGINAL division step1 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Valuation ORIGINAL a times b with exact rational arithmetic, not rounded decimals. - S020 · 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. - S019 · premise_and_result_audit — Recompute Valuation ORIGINAL b terminal factorization product with exact rational arithmetic, not rounded decimals. - S018 · 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. - S017 · premise_and_result_audit — Recompute Valuation ORIGINAL b dividing prime power4 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Valuation ORIGINAL b ORIGINAL division step4 with exact rational arithmetic, not rounded decimals. - S015 · 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. - S014 · premise_and_result_audit — Recompute Valuation ORIGINAL b dividing prime power3 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Valuation ORIGINAL b ORIGINAL division step3 with exact rational arithmetic, not rounded decimals. - S012 · 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. - S011 · premise_and_result_audit — Recompute Valuation ORIGINAL b dividing prime power2 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Valuation ORIGINAL b ORIGINAL division step2 with exact rational arithmetic, not rounded decimals. - S009 · 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. - S008 · premise_and_result_audit — Recompute Valuation ORIGINAL b dividing prime power1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Valuation ORIGINAL b ORIGINAL division step1 with exact rational arithmetic, not rounded decimals. - 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 (3, 24, -81, 1, 4, -1944, 5, 5, Yes, infinity_plus_k_equals_infinity) 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: cf46553e01feeafbf3209f4464187bd2fcfdc1efba614647b6e43b84f6c1f382 - candidate record SHA-256: fcc09bd702e1feed53f8a6e6619ba8cb5a34d9c13c3f313272f806d980e6a0f4 - 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-0033.json)