# Question 27478: Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations [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-0053) 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. R0053-27478 · q-20261015-advance-20261118-r05-p-adic-valuation-integers-0053 ## Original question and givens Use prime p=5, nonzero summands a=375, b=4; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed). - G001 — Use prime p=5, nonzero summands a=375, b=4; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed). ## 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-sum-ultrametric] After derived = [[p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed]] 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(375, 5) After Valuation ORIGINAL a ORIGINAL division step1 = 75 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, 5) After Valuation ORIGINAL a dividing prime power1 = 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, S002 S004 · impliesBefore source roles = [Valuation ORIGINAL a, 5, 375, 375, 1] After derived = [75, 5, 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 ORIGINAL division step2 := div(75, 5) After Valuation ORIGINAL a ORIGINAL division step2 = 15 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 · calculationBefore Valuation ORIGINAL a dividing prime power2 := mul(5, 5) After Valuation ORIGINAL a dividing prime power2 = 25 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, S005 S007 · impliesBefore source roles = [Valuation ORIGINAL a, 5, 375, 75, 2] After derived = [15, 25, 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, S006 S008 · calculationBefore Valuation ORIGINAL a ORIGINAL division step3 := div(15, 5) After Valuation ORIGINAL a 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, S007 S009 · calculationBefore Valuation ORIGINAL a dividing prime power3 := mul(25, 5) After Valuation ORIGINAL a dividing prime power3 = 125 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, S008 S010 · impliesBefore source roles = [Valuation ORIGINAL a, 5, 375, 15, 3] After derived = [3, 125, 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, S009 S011 · calculationBefore Valuation ORIGINAL a terminal factorization product := mul(125, 3) After Valuation ORIGINAL a terminal factorization product = 375 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 a, 5, 375, 375, 3, 3, [5, 25, 125]] After derived = [375, 3, 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, S011 S013 · calculationBefore Valuation ORIGINAL b terminal factorization product := mul(1, 4) After Valuation ORIGINAL b terminal factorization product = 4 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 · impliesBefore source roles = [Valuation ORIGINAL b, 5, 4, 4, 0, 4, []] After derived = [4, 4, 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, S013 S015 · calculationBefore Valuation ORIGINAL a plus b := sum(375, 4) After Valuation ORIGINAL a plus b = 379 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, S014 S016 · calculationBefore Valuation ACTUAL sum terminal factorization product := mul(1, 379) After Valuation ACTUAL sum terminal factorization product = 379 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 · impliesBefore source roles = [Valuation ACTUAL sum, 5, 379, 379, 0, 379, []] After derived = [379, 4, 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, S016 S018 · impliesBefore source roles = [5, 375, 4, 3, 0, 0] After derived = [379, 0, Yes, Yes, Yes] Conditions: none Factor the common p^min from the sum,so its valuation is at least min.If r