If $p$ and $q$ are distinct prime numbers, then there are integers $x_0$ and $y_0$ such that $1 = px_0+qy_0$. Determine the maximum value of $b-a$, where $a$ and $b$ are positive integers with the following property: If $a \leq t \leq b$, and $t$ is an integer, then there are integers $x$ and $y$ with $0 \leq x \leq q-1$ and $0 \leq y \leq p-1$ such that $t = px + qy$.
The first statement is obvious. It follows from Bezout's identity. Also if we want to determine the maximum value of $b-a$ it needs to be for all distinct primes $p,q$, so how do we ensure that?