To show that is an irrational number unless n is a perfect square, explain how the assump...

To show that is an irrational number unless n is a perfect square, explain how the assumption that is rational leads to a contradiction of the fundamental theorem of arithmetic by the following steps:

(A) Assume that n is not a perfect square, that is, does not belong to the sequence 1, 4, 9, 16, 25, … Explain why some prime number p appears an odd number of times as a factor in the prime factorization of n.

(B) Suppose that = a/b, where a and b are positive integers, b ≠ 0. Explain why a2 = nb2.

(C) Explain why the prime number p appears an even number of times (possibly 0 times) as a factor in the prime factorization of a2.

(D) Explain why the prime number p appears an odd number of times as a factor in the prime factorization of nb2.

(E) Explain why parts (C) and (D) contradict the fundamental theorem of arithmetic.