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The Fundamental Theorem of Arithmetic and Prime Factors

1. Prove the following lemma.

Lemma
Suppose that m and n are natural numbers > 1 and that p is a prime number.
The following statements are equivalent:

a. p is a prime factor of m or p is a prime factor of n.
b. p is a prime factor of m*n

Also Use Theorem:
The Fundamental Theorem of Arithmetic.
2. Prove the following corollary. You may use the lemma for this.

Corollary.
Suppose that n is a natural number > 1. Then the following statements are
equivalent:

a. p is a prime factor of n.
b. p is a prime factor of n^(1/2).

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