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    The Fundamental Theorem of Arithmetic and Prime Factors

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    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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    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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