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R0017-09072 · q-20261013-advance-20261018-r05-modular-congruence-0097
Use fixed positive modulus n=5, natural inputs a=22, b=27, c=45, d=55. 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=5, natural inputs a=22, b=27, c=45, d=55. 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=5
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=22;positive divisor=5
After
22=4·5+2
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=27;positive divisor=5
After
27=5·5+2
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=45;positive divisor=5
After
45=9·5+0
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=55;positive divisor=5
After
55=11·5+0
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(22, 45)
After
combined left value = 990
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(27, 55)
After
combined right value = 1485
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=990;positive divisor=5
After
990=198·5+0
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=1485;positive divisor=5
After
1485=297·5+0
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=[2, 2];c,d remainders=[0, 0]
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(4, 5)
After
first quotient difference = -1
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(9, 11)
After
second 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, S011
Before
product difference first term := mul(22, -2)
After
product difference first term = -44
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(55, -1)
After
product difference second term = -55
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(-44, -55)
After
product difference multiplier = -99
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(990, 1485)
After
combined difference = -495
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(5, -99)
After
modulus times difference multiplier = -495
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=-495;integer multiplier=-99;modulus=5
After
difference=n*multiplier=-495
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.(5, [22, 27], [45, 55], Yes, [0, 0], 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: b27973f7ec5c78fc9305ee16a48fab4183451510302a13aab8541f71ffa1c2e8candidate record SHA-256: 35c45a7e96448637dbb68f1a9b452166bd3f1d6d257a731cfcac51b94490c8easemantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED