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Let a, b, c, and m be positive integers. (a) Prove that the linear congruence ax = c (mod m) has either no solution or it has exactly ged(m, a) solutions in Zm-

Let a, b, c, and m be positive integers. (a) Prove that the linear congruence ax = c (mod m) has either no solution or it has exactly ged(m, a) solutions in Zm-

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let a, b, c, and m be positive integers. (a) Prove that the linear congruence ax = c (mod m) has either no solution or it has exactly gcd(m, a) solutions in Zm. (b) Prove that the linear congruence (with two variables) ax+by = c (mod m) has either no solution or it has exactly m · gcd(m, a, b) solutions (x, y) satisfying 0 ≤ x, y ≤ m − 1. (c) (not to be graded) Let a₁, a2, ..., ak be given positive integers. Prove that the linear congruence (with k variables) a₁x₁ + a₂x₂+...+akxk = c (mod m) •, *k) satisfying has either no solution or it has exactly mk-1.gcd(m, a₁, a2,, ak) solutions (x₁, x2, 0≤x1,x2,., xk ≤ m - 1.
Transcribed Image Text:Let a, b, c, and m be positive integers. (a) Prove that the linear congruence ax = c (mod m) has either no solution or it has exactly gcd(m, a) solutions in Zm. (b) Prove that the linear congruence (with two variables) ax+by = c (mod m) has either no solution or it has exactly m · gcd(m, a, b) solutions (x, y) satisfying 0 ≤ x, y ≤ m − 1. (c) (not to be graded) Let a₁, a2, ..., ak be given positive integers. Prove that the linear congruence (with k variables) a₁x₁ + a₂x₂+...+akxk = c (mod m) •, *k) satisfying has either no solution or it has exactly mk-1.gcd(m, a₁, a2,, ak) solutions (x₁, x2, 0≤x1,x2,., xk ≤ m - 1.
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