Question 395: A Nonunit Has No Modular Inverse

Original question and givens

Use the original signed integer a=21 and modulus n=42≥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. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.

• G001 — Use the original signed integer a=21 and modulus n=42≥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. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.

Definition and scope

When gcd(a,n)>1 the common divisor divides au+nv, so it cannot equal 1 and no inverse exists; null means absent.

• mko-modular-inverse

Step-by-step derivation

a=21; n=42≥2

compute gcd and a Bezout certificate before inversion

divide 21 by 42

21=0·42+21

divide 42 by 21

42=2·21+0

a·x := mul(21, 1)

a·x = 21

n·y := mul(42, 0)

n·y = 0

Bezout value := sum(21, 0)

Bezout value = 21

gcd(21,42)=21>1

no u can satisfy a·u≡1 (mod n)

inverse does not exist

inverse field=null; exists=false

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

(21, 42, 21, null, No, nonunit_has_no_modular_inverse)

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.