# The Fundamental Theorem of Arithmetic and Prime Factors

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See the attached file.

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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#### Solution Summary

The fundamental theorem of arithmetic and prime factors are investigated and the details are discussed. The solution is detailed and well presented.

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