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R0053-27446 · q-20261015-advance-20261118-r05-p-adic-valuation-integers-0021
Use prime p=5, nonzero integer a=875. 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=875. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).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.
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.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
Valuation ORIGINAL a ORIGINAL division step1 := div(875, 5)
After
Valuation ORIGINAL a ORIGINAL division step1 = 175
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S001
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.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
source roles = [Valuation ORIGINAL a, 5, 875, 875, 1]
After
derived = [175, 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.
Rule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S003
Before
Valuation ORIGINAL a ORIGINAL division step2 := div(175, 5)
After
Valuation ORIGINAL a ORIGINAL division step2 = 35
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S004
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.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
source roles = [Valuation ORIGINAL a, 5, 875, 175, 2]
After
derived = [35, 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.
Rule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S006
Before
Valuation ORIGINAL a ORIGINAL division step3 := div(35, 5)
After
Valuation ORIGINAL a ORIGINAL division step3 = 7
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S007
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.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
source roles = [Valuation ORIGINAL a, 5, 875, 35, 3]
After
derived = [7, 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.
Rule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S009
Before
Valuation ORIGINAL a terminal factorization product := mul(125, 7)
After
Valuation ORIGINAL a terminal factorization product = 875
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
source roles = [Valuation ORIGINAL a, 5, 875, 875, 3, 7, [5, 25, 125]]
After
derived = [875, 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.
Rule: Valuation_terminal_NONdivisible_unit_highest_power · Uses: G001, S011
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.(5, 875, 875, 3, 7, [5, 25, 125], Yes, Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
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