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Gulf Mathematical Olympiad 2013 - Problem 4

Source: Gulf Mathematical Olympiad 2013

April 5, 2013
number theorygreatest common divisornumber theory unsolved

Problem Statement

Let m,nm,n be integers. It is known that there are integers a,ba,b such that am+bn=1am+bn=1 if, and only if, the greatest common divisor of m,nm,n is 1. You are not required to prove this.
Now suppose that p,qp,q are different odd primes. In each case determine if there are integers a,ba,b such that ap+bq=1ap+bq=1 so that the given condition is satisfied:
a. pp divides bb and qq divides aa; b. pp divides aa and qq divides bb; c. pp does not divide aa and qq does not divide bb.
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