Hint for Section 3.6 Question 1

1. We are asked to prove, using the method of contradiction, the statement: "n Z and "primes p,  if p | n², then p | n.

The negation of the given statement is:  $ n Z and $ a prime p such that  p | n²  and  pn.

We now assume that the negation is true and we need to show that this leads to a contradiction.

Consider the prime factorization (from the Unique Factorization Theorem) of each of n and n². 

If the unique prime factorization of n is:   n = p₁^e₁ · p₂^e₂ · ... · p_k^e_k,  then what can you say about the unique prime factorization of n²? Next consider what
p | n²  and pn  mean in terms of the unique prime factorizations.

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