Determine if the following is a valid argument. Explain your conclusion.Let m and n be two...

Determine if the following is a valid argument. Explain your conclusion.

Let m and n be two relatively prime integers. Prove that if mn is a cube, then m and n are each cubes.

Proof: We first note that in the factorization of any cube into prime factors, each prime must have an exponent that is a multiple of 3. Write m and n each as a product of primes;  Suppose m is not a cube. Then at least one ai is not a multiple of 3. Since each prime factor of mn must have an exponent that is a multiple of 3, n must have a factor  such that bi ≠ 0 and ai + bi is a multiple of 3. But this means that m and n share a factor, pi. This contradicts the fact that m and n are relatively prime.