Question 365: Keep Signed Bezout Coefficients and Verify the Integer Identity

Original question and givens

Use the original signed integer a=2 and modulus n=41≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.

• G001 — Use the original signed integer a=2 and modulus n=41≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.

Definition and scope

Run extended Euclid on |a| and n and restore the sign of a. The coefficients are one Bezout witness, not a unique pair.

• mko-modular-inverse

Step-by-step derivation

a=2; n=41≥2

compute gcd and a Bezout certificate before inversion

divide 2 by 41

2=0·41+2

divide 41 by 2

41=20·2+1

divide 2 by 1

2=2·1+0

a·x := mul(2, -20)

a·x = -40

n·y := mul(41, 1)

n·y = 41

Bezout value := sum(-40, 41)

Bezout value = 1

2·(-20)+41·(1)=1

inverse representative=21; 2·21≡1 (mod 41)

Reverse checks

• S008 · premise_and_result_audit — Check integer identities, remainder bounds and gcd conditions; congruence is not integer equality.
• S007 · premise_and_result_audit — Recompute Bezout value with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute n·y with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute a·x with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Check integer identities, remainder bounds and gcd conditions; congruence is not integer equality.
• S003 · premise_and_result_audit — Check integer identities, remainder bounds and gcd conditions; congruence is not integer equality.
• S002 · premise_and_result_audit — Check integer identities, remainder bounds and gcd conditions; congruence is not integer equality.
• S001 · premise_and_result_audit — Check integer identities, remainder bounds and gcd conditions; congruence is not integer equality.

Result

(2, 41, 1, -20, 1, 1, Bezout_equals_one)

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.

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.