(c) Prove that m is prime if and only if y(m) = m - 1.

For any integer k greater than or equal to 2, define R(k) as the ratio of k 's largest positive prime divisor to its smallest positive prime divisor. For instance, R(14) = ½, R(15) = 3, R(30) = ½, and R(9) = 3 = 1. Note that R(k) = 1 if k is a prime number or any power of a prime. Additionally, R(1) is defined as 1. Prove or disprove these statements: (a) For every natural number x, there is a natural number y such that the ratio R(y) is at least x. (b) There is a natural number y such that for every natural number x, the ratio R(y) is at least x.

Prove that (z) = (zk) for every integer k (provided z 0 when k is negative).

By contradiction