smc

We have $y=\frac{501-3x}{5}$. Thus, $501-3x$ must be a positive multiple of $5$. If $x=2$, we find our first positive multiple of $5$. From there, we note that $x=2+5k$ will always return a multiple of $5$ for $501-3x$. Our first solution happens at $k=0$. We now want to find the smallest multiple of $5$ that will work. If $x=2+5k$, then we have $501-3x=501-3(2+5k)$, or $495-15k$. When $k=32$, the expression is equal to $15$, and when $k=33$, the expression is equal to $0$, which will no longer work. Thus, all integers from $k=0$ to $k=32$ will generate an $x=2+5k$ that will be a positive integer, and which will in turn generate a $y$ that is also a positive integer. So, the answer is $\boxed{33}$.