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

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As $501-3x=3(167-x)$, the given equation is equivalent to $y=\frac{3}{5}(167-x)$. Since $y$ is a positive integer, $167-x$ must be some positive multiple of 5, that is, $167-x=5k$, for some positive integer $k$ and, therefore, $x=167-5k$ or, still, $x=5 \times 33+2-5k=5(33-k)+2$. Since $x$ is a positive integer, we must have $1 \leq k \leq 33$. Consequently, we can take any $k=1,2, \ldots, 33$, obtaining thirty-three pairs of positive integers $(x, y)$ that are solutions to the equation $3x+57=501$.