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Question 27473: Integer Valuations Add under Products, Including the Fixed Zero Convention

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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R0053-27473 · q-20261015-advance-20261118-r05-p-adic-valuation-integers-0048

Original question and givens

Use prime p=11, integers a=253, b=-2904, 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=11, integers a=253, b=-2904, 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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

Valuation ORIGINAL a ORIGINAL division step1 := div(253, 11)

After

Valuation ORIGINAL a ORIGINAL division step1 = 23

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, 11)

After

Valuation ORIGINAL a dividing prime power1 = 11

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, 11, 253, 253, 1]

After

derived = [23, 11, 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 terminal factorization product := mul(11, 23)

After

Valuation ORIGINAL a terminal factorization product = 253

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 · implies

Before

source roles = [Valuation ORIGINAL a, 11, 253, 253, 1, 23, [11]]

After

derived = [253, 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, S005

S007 · calculation

Before

Valuation ORIGINAL b ORIGINAL division step1 := div(2904, 11)

After

Valuation ORIGINAL b ORIGINAL division step1 = 264

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, S006

S008 · calculation

Before

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

After

Valuation ORIGINAL b dividing prime power1 = 11

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 b, 11, 2904, 2904, 1]

After

derived = [264, 11, 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, S008

S010 · calculation

Before

Valuation ORIGINAL b ORIGINAL division step2 := div(264, 11)

After

Valuation ORIGINAL b ORIGINAL division step2 = 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, S009

S011 · calculation

Before

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

After

Valuation ORIGINAL b dividing prime power2 = 121

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, 11, 2904, 264, 2]

After

derived = [24, 121, 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 terminal factorization product := mul(121, 24)

After

Valuation ORIGINAL b terminal factorization product = 2904

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 · implies

Before

source roles = [Valuation ORIGINAL b, 11, -2904, 2904, 2, 24, [11, 121]]

After

derived = [2904, 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, S013

S015 · calculation

Before

Valuation ORIGINAL a times b := mul(253, -2904)

After

Valuation ORIGINAL a times b = -734712

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, S014

S016 · calculation

Before

Valuation ACTUAL product ORIGINAL division step1 := div(734712, 11)

After

Valuation ACTUAL product ORIGINAL division step1 = 66792

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

After

Valuation ACTUAL product dividing prime power1 = 11

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 ACTUAL product, 11, 734712, 734712, 1]

After

derived = [66792, 11, 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 ACTUAL product ORIGINAL division step2 := div(66792, 11)

After

Valuation ACTUAL product ORIGINAL division step2 = 6072

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

After

Valuation ACTUAL product dividing prime power2 = 121

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 ACTUAL product, 11, 734712, 66792, 2]

After

derived = [6072, 121, 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 ACTUAL product ORIGINAL division step3 := div(6072, 11)

After

Valuation ACTUAL product ORIGINAL division step3 = 552

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 · calculation

Before

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

After

Valuation ACTUAL product dividing prime power3 = 1331

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, S022

S024 · implies

Before

source roles = [Valuation ACTUAL product, 11, 734712, 6072, 3]

After

derived = [552, 1331, 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, S023

S025 · calculation

Before

Valuation ACTUAL product terminal factorization product := mul(1331, 552)

After

Valuation ACTUAL product terminal factorization product = 734712

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 · implies

Before

source roles = [Valuation ACTUAL product, 11, -734712, 734712, 3, 552, [11, 121, 1331]]

After

derived = [734712, 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, S025

S027 · calculation

Before

Valuation ORIGINAL exponent addition := sum(1, 2)

After

Valuation ORIGINAL exponent addition = 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, S026

S028 · implies

Before

source roles = [11, 253, -2904, 1, 2, 23, 24]

After

derived = [-734712, 3, 3, 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 dependencies

Rule: Valuation_GENERAL_product_primeness_unit_product_or_zero · Uses: G001, S027

Reverse checks

  • S028 · 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.
  • S027 · premise_and_result_audit — Recompute Valuation ORIGINAL exponent addition with exact rational arithmetic, not rounded decimals.
  • S026 · 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.
  • S025 · premise_and_result_audit — Recompute Valuation ACTUAL product terminal factorization product 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 power3 with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Recompute Valuation ACTUAL product ORIGINAL division step3 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 ACTUAL product dividing prime power2 with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Recompute Valuation ACTUAL product ORIGINAL division step2 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 ACTUAL product dividing prime power1 with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute Valuation ACTUAL product ORIGINAL division step1 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Valuation ORIGINAL a times b with exact rational arithmetic, not rounded decimals.
  • S014 · 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.
  • S013 · premise_and_result_audit — Recompute Valuation ORIGINAL b terminal factorization product 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

(11, 253, -2904, 1, 2, -734712, 3, 3, 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: 83b8515bff8be18010894e429c7e8efd01a0871d9e1b15cf6c92922c096f35d1
  • candidate record SHA-256: bbd12fa774868a3ffa8ec614802bd5ef8123ecdc665bd5fbd15d2a79a07afdc4
  • semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62
  • question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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