GULF 2013 Problem 4

Let $m,n$ be integers. It is known that there are integers $a,b$ such that $am+bn=1$ if, and only if, the greatest common divisor of $m,n$ is 1. You are not required to prove this.

Now suppose that $p,q$ are different odd primes. In each case determine if there are integers $a,b$ such that $ap+bq=1$ so that the given condition is satisfied: [list] a. $p$ divides $b$ and $q$ divides $a$ ; b. $p$ divides $a$ and $q$ divides $b$ ; c. $p$ does not divide $a$ and $q$ does not divide $b$ . [/list]