Bezout’s Theorem and the Fundamental Theorem of Arithmetic 1. Let a, b, c ∈ Z. Prove...

Bezout’s Theorem and the Fundamental Theorem of Arithmetic

1. Let a, b, c ∈ Z. Prove that c = ma + nb for some m, n ∈ Z if and only if gcd(a, b)|c.

2. Prove that if c|ab and gcd(a, c) = 1, then c|b.

3. Prove that for all a, b ∈ Z not both zero, gcd(a, b) = 1 if and only if a and b have no prime factors in common.

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Expert Solution

Colby Messinger answered 1 year ago

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