S001 · implies
Before
a=5, modulus=42
After
n=42≥2; seek u with 5u≡1 (mod 42)
Conditions: none
Fix the modulus and inverse condition; the integer product need not equal 1.
Step identifiers and dependencies
Rule: inverse_definition · Uses: G001
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Question 66: Normalize the Inverse Representative, Not the Bezout Coefficients
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P250-0066 · q-20261014-advance-20261103-r07-modular-inverse-0066
Original question and givens
Use the original signed integer a=5 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,inverse,product_mod_n,valid?,unique_modulo_n).
G001— Use the original signed integer a=5 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,inverse,product_mod_n,valid?,unique_modulo_n).
Definition and scope
Use integer modulus n≥2 with gcd(a,n)=1. Uniqueness concerns the residue class and its representative in the declared interval.
Step-by-step derivation
S002 · implies
Before
divide 5 by 42
After
5=0·42+5
Conditions: 0≤5<42
The quotient is 0 and remainder 5; continue with the preceding divisor and remainder.
Step identifiers and dependencies
Rule: euclidean_division · Uses: G001, S001
S003 · implies
Before
divide 42 by 5
After
42=8·5+2
Conditions: 0≤2<5
The quotient is 8 and remainder 2; continue with the preceding divisor and remainder.
Step identifiers and dependencies
Rule: euclidean_division · Uses: G001, S002
S004 · implies
Before
divide 5 by 2
After
5=2·2+1
Conditions: 0≤1<2
The quotient is 2 and remainder 1; continue with the preceding divisor and remainder.
Step identifiers and dependencies
Rule: euclidean_division · Uses: G001, S003
S005 · implies
Before
divide 2 by 1
After
2=2·1+0
Conditions: 0≤0<1
The quotient is 2 and remainder 0; continue with the preceding divisor and remainder.
Step identifiers and dependencies
Rule: euclidean_division · Uses: G001, S004
S006 · implies
Before
last nonzero Euclidean remainder
After
gcd(5,42)=1; 5·(17)+42·(-2)=1
Conditions: none
Track integer linear combinations along the division chain to obtain this explicit Bezout identity.
Step identifiers and dependencies
Rule: bezout_identity · Uses: G001, S005
S007 · implies
Before
5·(17)+42·(-2)=1
After
17 mod 42=17; 0≤17<42
Conditions: none
The coefficient of a represents the inverse class; reduce it to the required representative interval.
Step identifiers and dependencies
Rule: normalize_residue · Uses: G001, S006
S008 · calculation
Before
a·u := mul(5, 17)
After
a·u = 85
Conditions: none
Multiply the original a by the normalized inverse.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · implies
Before
a·u=85
After
85=2·42+1 ⇒ a·u mod n=1
Conditions: none
The product minus 1 is an integer multiple of the modulus, verifying the inverse condition.
Step identifiers and dependencies
Rule: congruence_check · Uses: G001, S008
S010 · implies
Before
a·v≡a·u≡1 (mod n), gcd(a,n)=1
After
n divides v−u; if 0≤u,v<n then u=v
Conditions: none
From n|a(v−u), multiply the Bezout identity by v−u to get n|(v−u). Two normalized representatives differ by less than n in absolute value, so their difference is zero.
Step identifiers and dependencies
Rule: inverse_uniqueness · Uses: G001, S009
Reverse checks
S010· premise_and_result_audit — The class and its normalized representative are unique; u+kn gives other integer representatives.S009· premise_and_result_audit — The quotient is an integer and the remainder really is 1.S008· premise_and_result_audit — Recompute a·u with exact rational arithmetic, not rounded decimals.S007· premise_and_result_audit — Normalize the inverse representative, not both Bezout coefficients.S006· premise_and_result_audit — Multiply out the identity directly rather than trusting a gcd label.S005· premise_and_result_audit — Check both the division identity and 0≤remainder<divisor.S004· premise_and_result_audit — Check both the division identity and 0≤remainder<divisor.S003· premise_and_result_audit — Check both the division identity and 0≤remainder<divisor.S002· premise_and_result_audit — Check both the division identity and 0≤remainder<divisor.S001· premise_and_result_audit — A modular inverse requires gcd(a,n)=1.
Result
(5, 42, 17, 1, Yes, unique_modulo_n)
The inverse representative is 17, unique modulo 42.
This question review and source binding
Review date: 2026-10-24
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
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