See the attached file.
1. Prove the following 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.
Suppose that n is a natural number > 1. Then the following statements are
a. p is a prime factor of n.
b. p is a prime factor of n^(1/2).
The fundamental theorem of arithmetic and prime factors are investigated and the details are discussed. The solution is detailed and well presented.