OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 27440: The Highest Dividing Prime Power Is Extracted from the Absolute Integer

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

Return to catalogue page 138

← Question derivation index · Original question batch

R0053-27440 · q-20261015-advance-20261118-r05-p-adic-valuation-integers-0015

Original question and givens

Use prime p=5, nonzero integer a=125. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

  • G001 — Use prime p=5, nonzero integer a=125. 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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

Valuation ORIGINAL a ORIGINAL division step1 := div(125, 5)

After

Valuation ORIGINAL a ORIGINAL division step1 = 25

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, 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

source roles = [Valuation ORIGINAL a, 5, 125, 125, 1]

After

derived = [25, 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 dependencies

Rule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S003

S005 · calculation

Before

Valuation ORIGINAL a ORIGINAL division step2 := div(25, 5)

After

Valuation ORIGINAL a ORIGINAL division step2 = 5

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(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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

source roles = [Valuation ORIGINAL a, 5, 125, 25, 2]

After

derived = [5, 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 dependencies

Rule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S006

S008 · calculation

Before

Valuation ORIGINAL a ORIGINAL division step3 := div(5, 5)

After

Valuation ORIGINAL a ORIGINAL division step3 = 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, S007

S009 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

source roles = [Valuation ORIGINAL a, 5, 125, 5, 3]

After

derived = [1, 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 dependencies

Rule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S009

S011 · calculation

Before

Valuation ORIGINAL a terminal factorization product := mul(125, 1)

After

Valuation ORIGINAL a terminal factorization product = 125

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 a, 5, 125, 125, 3, 1, [5, 25, 125]]

After

derived = [125, 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, S011

Reverse checks

  • 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 a terminal factorization product with exact rational arithmetic, not rounded decimals.
  • S010 · 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.
  • S009 · premise_and_result_audit — Recompute Valuation ORIGINAL a dividing prime power3 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Valuation ORIGINAL a ORIGINAL division step3 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

(5, 125, 125, 3, 1, [5, 25, 125], 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: b2ebfdeb4fea4fa902747b3659f9e79865947ec30433c6984b85ed819367f114
  • candidate record SHA-256: 1c8ba81afc2240cebb053253fe88fe30a3bafb2f169db6c128c4a61cc03d6f9a
  • semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62
  • question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON