Assignment:
Recall that a perfect square is a natural number n such that n = (k^2), for some natural number k.
Theorem. If the natural number n is not a perfect square, then n^(1/2) is irrational.
Proof.
S(1): Suppose n^(1/2) = r/s for some natural numbers r and s.
S(2): We may assume that r and s have no prime factors in common, since any common prime factors may be cancelled.
S(3): From the first step, we have (s^2)*n = r^2.
S(4): Suppose that s>1 and p is a prime factor of s.
S(5): Then p is a prime factor of s^2.
S(6): Hence p is a prime factor of r^2 = (s^2)*n.
S(7): It follows that p is a prime factor of r.
S(8): This contradicts our assumption that r and s have no prime factors in common, and so s = 1.
S(9): Therefore, n = (r^2), so n is a perfect square.
Explain all S(1) ~ S(9)
Provide complete and step by step solution for the question and show calculations and use formulas.