We want to find conditions on a, b, and c such that gcd(a, c)gcd(b, c) = gcd(ab, c). (a) Find a counterexample to the statement gcd(a, c)gcd(b, c) = gcd(ab, c). This means that some conditions have to be placed on a, b, and c. (b) Prove that if gcd(a, c) = gcd(b, c) = 1, then gcd(ab, c) = 1. (c) Prove that if gcd(a, c) = 1, then gcd(b, c) = gcd(ab, c). (d) Prove that if gcd(a, b) = 1, then gcd(a, c)gcd(b, c) = gcd(ab, c). (e) (bonus) Prove that if gcd(a, b, c) = 1, then gcd(a, c)gcd(b, c) = gcd(ab, c). Is this an if and only if condition?

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A: Solution is given below:

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A: To Show: U(8) is not isomorphic to U(10)

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