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R0017-09036 · q-20261013-advance-20261018-r05-modular-congruence-0036
Fix positive modulus n=3, natural a=31, bound=35. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
G001 — Fix positive modulus n=3, natural a=31, bound=35. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).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=3
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
representative remainder:natural dividend=31;positive divisor=3
After
31=10·3+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, S001
Before
window bound minus residue := sub(35, 1)
After
window bound minus residue = 34
Conditions: none
Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
last progression index:natural dividend=34;positive divisor=3
After
34=11·3+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, S003
Before
residue=1;modulus=3;0<=k<=11
After
all class members=[1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34]
Conditions: none
Euclidean uniqueness gives x≡a exactly when x=r+k*n.Natural x and0≤r<n force k≥0;the bound gives k≤floor((bound−r)/n).Retain the full inclusive progression,not a sample.
Rule: complete_residue_progression · Uses: G001, S004
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 — Recompute window bound minus residue with exact rational arithmetic, not rounded decimals.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.(3, 31, 35, 1, [1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34])
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: d3a195c8daaefe4a590c86e178443c257a95839aa1fae7aa8751eddecfa2942ccandidate record SHA-256: 11159cb641a40cc45124e1e594fd84facc986bdc884927b58122f9041be0e3f9semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED