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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