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R0017-09067 · q-20261013-advance-20261018-r05-modular-congruence-0092
Use fixed positive modulus n=9, natural inputs a=17, b=35, c=35, d=62. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
G001 — Use fixed positive modulus n=9, natural inputs a=17, b=35, c=35, d=62. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.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
fixed natural modulus candidate=9
After
positive modulus=Yes
Conditions: none
Congruence compares remainders at one positive modulus,not numerical equality.Multiplicative preservation does not permit unconditional cancellation of a common factor.
Rule: fixed_positive_modulus_scope · Uses: G001
Before
a remainder:natural dividend=17;positive divisor=9
After
17=1·9+8
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S001
Before
b remainder:natural dividend=35;positive divisor=9
After
35=3·9+8
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S002
Before
c remainder:natural dividend=35;positive divisor=9
After
35=3·9+8
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S003
Before
d remainder:natural dividend=62;positive divisor=9
After
62=6·9+8
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S004
Before
combined left value := mul(17, 35)
After
combined left value = 595
Conditions: none
Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
combined right value := mul(35, 62)
After
combined right value = 2170
Conditions: none
Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
combined left remainder:natural dividend=595;positive divisor=9
After
595=66·9+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S007
Before
combined right remainder:natural dividend=2170;positive divisor=9
After
2170=241·9+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S008
Before
a,b remainders=[8, 8];c,d remainders=[8, 8]
After
both premises=Yes;conclusion=Yes
Conditions: none
Audit both premises first,then independently compute the sum/product remainders.A failed premise does not force a false conclusion.
Rule: modular_premises_and_conclusion_separate · Uses: G001, S009
Before
first quotient difference := sub(1, 3)
After
first quotient difference = -2
Conditions: none
Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
second quotient difference := sub(3, 6)
After
second quotient difference = -3
Conditions: none
Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
product difference first term := mul(17, -3)
After
product difference first term = -51
Conditions: none
Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
product difference second term := mul(62, -2)
After
product difference second term = -124
Conditions: none
Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
product difference multiplier := sum(-51, -124)
After
product difference multiplier = -175
Conditions: none
Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
combined difference := sub(595, 2170)
After
combined difference = -1575
Conditions: none
Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
modulus times difference multiplier := mul(9, -175)
After
modulus times difference multiplier = -1575
Conditions: none
Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
combined difference=-1575;integer multiplier=-175;modulus=9
After
difference=n*multiplier=-1575
Conditions: none
Equal remainders give a−b=n(qa−qb) and c−d=n(qc−qd).Add directly,or use ac−bd=a(c−d)+d(a−b).The multiplier may be a negative integer;no factor cancellation is needed.
Rule: applicable_modular_preservation_witness · Uses: G001, S017
Before
premises=Yes;conclusion=Yes
After
material implication=Yes
Conditions: none
A false premise makes the material implication true by its truth definition,not by supplying applicable preservation premises.
Rule: modular_implication_not_applicable_premise · Uses: G001, S018
S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S017 · premise_and_result_audit — Recompute modulus times difference multiplier with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute combined difference with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute product difference multiplier with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute product difference second term with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute product difference first term with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute second quotient difference with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute first quotient difference with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S007 · premise_and_result_audit — Recompute combined right value with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute combined left value with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(9, [17, 35], [35, 62], Yes, [1, 1], 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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 2842ae496d58deed39d5c5dbb2d3a8545bd659fbb4350ec26a0d1002971fc6c9candidate record SHA-256: f785967694ac65812c73e3e7a7914fb355916d4bb5dc83ebd6bcc9d254c419easemantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED