Let a, b, c be positive integers. Prove that there exists integers x, y, z such that ax + by + cz = gcd(a; b; c). Give the names of any theorems you are using. Hint: Substitute gcd(a, b) = am+bn into gcd(a, b)r + cs = gcd(gcd(a,b),c) = gcd(a, b, c) and multiply out the result.

Let a, b, c be positive integers. Prove that there exists integers x, y, z such that ax + by + cz = gcd(a; b; c). Give the names of any theorems you are using. Hint: Substitute gcd(a, b) = am+bn into gcd(a, b)r + cs = gcd(gcd(a,b),c) = gcd(a, b, c) and multiply out the result.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.2: Mappings

Problem 23E: Let a and b be constant integers with a0, and let the mapping f:ZZ be defined by f(x)=ax+b. Prove...

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Let a, b, c be positive integers. Prove that there exists integers x, y, z such that ax + by + cz = gcd(a; b; c). Give the names of any theorems you are using. Hint: Substitute gcd(a, b) = am+bn into gcd(a, b)r + cs = gcd(gcd(a,b),c) = gcd(a, b, c) and multiply out the result.

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