Use the complete prime field Z_13, with prime p=13, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=6, b=6, c=11; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
G001 — Use the complete prime field Z_13, with prime p=13, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=6, b=6, c=11; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
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.
THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged.
Trial EVERY divisor2 through floor(sqrt(p));a composite would have a factor there.The empty test for p=2 is valid.This qualifies THIS prime field Z_p,not every finite field’s order or every finite ring as a field.
Field ORIGINAL a times proposed inverse := mul(6, 11)
After
Field ORIGINAL a times proposed inverse = 66
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Field ORIGINAL inverse times c minus b := mul(11, 5)
After
Field ORIGINAL inverse times c minus b = 55
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Prime p and nonzero a give an inverse.Subtract b,then multiply by it and substitute under the SAME mod-p operations.If x,z solve,then a(x−z)=0;inverse multiplication gives x=z,proving uniqueness.Never divide by0.
S011 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
S010 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
S009 · premise_and_result_audit — Recompute Field ORIGINAL a solution plus b with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Recompute Field ORIGINAL a times solution with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
S006 · premise_and_result_audit — Recompute Field ORIGINAL inverse times c minus b with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute Field ORIGINAL c minus b with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
S003 · premise_and_result_audit — Recompute Field ORIGINAL a times proposed inverse with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
S001 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
Result
(6, 6, 11, 11, 3, 11, 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.