How many pairs of positive integers $(x, y)$ are solutions to the equation $3 x+5 y=501 ?$

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The given equation is equivalent to $y=\frac{3(167-x)}{5}$. Since $y$ is a positive integer, $167-x$ must be a positive multiple of 5, that is:

$167-x=5 k \quad \Rightarrow \quad x=167-5 k \quad \Rightarrow \quad x=5 \times 33+2-5 k \quad \Rightarrow \quad x=5(33-k)+2$

where $k$ is a positive integer. Since $x$ is positive, we must have $k \leq 33$. Consequently, $k=1,2, \ldots, 33$ which guarantees 33 solutions to the proposed problem.