DE-BUNDESWETTBEWERB 2003 Problem 8

Let $p$ and $q$ be two positive integers that have no common divisor. The set of integers shall be partioned into three subsets $A$ , $B$ , $C$ such that for each integer $z$ in each of the sets $A$ , $B$ , $C$ there is exactly one of the numbers $z$ , $z+p$ and $z+q$ .

a) Prove that such a decomposition is possible if and only if $p+q$ is divisible by $3$ .

b) In the case we omit the restriction that $p$ , $q$ may not have a common divisor, prove that for $p \neq q$ the number $\frac{p+q}{\gcd(p,q)}$ is divisible by 3.