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ASU 068 All Russian MO 1965 n = px + qy, relative primes p,q

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June 19, 2019
number theoryrelatively prime

Problem Statement

Given two relatively prime numbers p>0p>0 and q>0q>0. An integer nn is called "good" if we can represent it as n=px+qyn = px + qy with nonnegative integers xx and yy, and "bad" in the opposite case.
a) Prove that there exist integer cc such that in a pair {n,cāˆ’n}\{n, c-n\} always one is "good" and one is "bad".
b) How many there exist "bad" numbers?
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