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Question 27483: Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0053-27483 · q-20261015-advance-20261118-r05-p-adic-valuation-integers-0058

Original question and givens

Use prime p=3, nonzero summands a=72, b=81; 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=3, nonzero summands a=72, b=81; 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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

Valuation ORIGINAL a ORIGINAL division step1 := div(72, 3)

After

Valuation ORIGINAL a ORIGINAL division step1 = 24

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

source roles = [Valuation ORIGINAL a, 3, 72, 72, 1]

After

derived = [24, 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 dependencies

Rule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S003

S005 · calculation

Before

Valuation ORIGINAL a ORIGINAL division step2 := div(24, 3)

After

Valuation ORIGINAL a ORIGINAL division step2 = 8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Valuation ORIGINAL a dividing prime power2 := mul(3, 3)

After

Valuation ORIGINAL a 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

source roles = [Valuation ORIGINAL a, 3, 72, 24, 2]

After

derived = [8, 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 dependencies

Rule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S006

S008 · calculation

Before

Valuation ORIGINAL a terminal factorization product := mul(9, 8)

After

Valuation ORIGINAL a terminal factorization product = 72

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · implies

Before

source roles = [Valuation ORIGINAL a, 3, 72, 72, 2, 8, [3, 9]]

After

derived = [72, 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 dependencies

Rule: Valuation_terminal_NONdivisible_unit_highest_power · Uses: G001, S008

S010 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

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 dependencies

Rule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S011

S013 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · implies

Before

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 dependencies

Rule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S014

S016 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

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 dependencies

Rule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S017

S019 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · implies

Before

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 dependencies

Rule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S020

S022 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · implies

Before

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 dependencies

Rule: Valuation_terminal_NONdivisible_unit_highest_power · Uses: G001, S022

S024 · calculation

Before

Valuation ORIGINAL a plus b := sum(72, 81)

After

Valuation ORIGINAL a plus b = 153

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

Valuation ACTUAL sum ORIGINAL division step1 := div(153, 3)

After

Valuation ACTUAL sum ORIGINAL division step1 = 51

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

Valuation ACTUAL sum dividing prime power1 := mul(1, 3)

After

Valuation ACTUAL sum 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · implies

Before

source roles = [Valuation ACTUAL sum, 3, 153, 153, 1]

After

derived = [51, 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 dependencies

Rule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S026

S028 · calculation

Before

Valuation ACTUAL sum ORIGINAL division step2 := div(51, 3)

After

Valuation ACTUAL sum ORIGINAL division step2 = 17

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

Valuation ACTUAL sum dividing prime power2 := mul(3, 3)

After

Valuation ACTUAL sum 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · implies

Before

source roles = [Valuation ACTUAL sum, 3, 153, 51, 2]

After

derived = [17, 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 dependencies

Rule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S029

S031 · calculation

Before

Valuation ACTUAL sum terminal factorization product := mul(9, 17)

After

Valuation ACTUAL sum terminal factorization product = 153

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · implies

Before

source roles = [Valuation ACTUAL sum, 3, 153, 153, 2, 17, [3, 9]]

After

derived = [153, 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 dependencies

Rule: Valuation_terminal_NONdivisible_unit_highest_power · Uses: G001, S031

S033 · implies

Before

source roles = [3, 72, 81, 2, 4, 2]

After

derived = [153, 2, Yes, Yes, Yes]

Conditions: none

Factor the common p^min from the sum,so its valuation is at least min.If r<s,the bracket u+p^(s-r)v is congruent to nonzero u mod p,hence has exact exponent r;the symmetric case is analogous.When r=s,cancellation may raise the exponent or equality may remain.The last field is not_guaranteed,not a claim that this instance fails equality.

Step identifiers and dependencies

Rule: Valuation_GENERAL_common_power_ultrametric_AND_unequal_case · Uses: G001, S032

Reverse checks

  • 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 — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S031 · premise_and_result_audit — Recompute Valuation ACTUAL sum terminal factorization product 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 sum dividing prime power2 with exact rational arithmetic, not rounded decimals.
  • S028 · premise_and_result_audit — Recompute Valuation ACTUAL sum ORIGINAL division step2 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 sum dividing prime power1 with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Recompute Valuation ACTUAL sum ORIGINAL division step1 with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Recompute Valuation ORIGINAL a plus b with exact rational arithmetic, not rounded decimals.
  • S023 · 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.
  • S022 · premise_and_result_audit — Recompute Valuation ORIGINAL b terminal factorization product with exact rational arithmetic, not rounded decimals.
  • S021 · 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.
  • S020 · premise_and_result_audit — Recompute Valuation ORIGINAL b dividing prime power4 with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Recompute Valuation ORIGINAL b ORIGINAL division step4 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 power3 with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute Valuation ORIGINAL b ORIGINAL division step3 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 power2 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute Valuation ORIGINAL b ORIGINAL division step2 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 power1 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Valuation ORIGINAL b ORIGINAL division step1 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 a terminal factorization product with exact rational arithmetic, not rounded decimals.
  • S007 · 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.
  • S006 · premise_and_result_audit — Recompute Valuation ORIGINAL a dividing prime power2 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Valuation ORIGINAL a ORIGINAL division step2 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, 72, 81, 2, 4, 153, 2, 2, Yes, 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: ab7993b1a8c5832d0fb93b1725ad81ec765fe8302ea962442be004d8917aec98
  • candidate record SHA-256: 06e0d33e7e1557d18f96d8a4e7c782c74d0f3177263440ad44b1369ba8556be6
  • semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62
  • question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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