Prove that the set of positive rational numbers is countable by showing that the function K is a one-to-one correspondence between the set of positive rational numbers and the set of positive integers if K(m/n) = p12a1 p22a2 · · · · · ps2as q12b1−1 q22b2−1 · · · · · qt2bt−1, where gcd(m, n) = 1 and the prime-power factorizations of m and n are m = p1a1 p2a2 · · · · · psas and n = q1b1 q2b2 · · · qtbt.