3. If $x, y$ are two coprime positive integers, and
$$
x+\frac{2016}{x}=32 y+\frac{63}{y} \text {, }
$$

then the number of pairs $(x, y)$ that satisfy the condition is $(\quad)$.
(A) 1
(B) 4
(C) 5
(D) infinitely many
3. C.

From the given equation, we get $(x-32 y)\left(1-\frac{63}{x y}\right)=0$. Therefore, $x-32 y=0$ or $1-\frac{63}{x y}=0$.
When $x-32 y=0$, i.e., $x=32 y$, combining with $(x, y)$ $=1$, only $(x, y)=(32,1)$ satisfies the requirement;

When $1-\frac{63}{x y}=0$, i.e., $x y=63$, combining with $(x, y)$ $=1$, it is easy to see,
$$
(x, y)=(63,1),(9,7),(7,9),(1,63) \text {. }
$$

In summary, there are 5 pairs of numbers that meet the requirements.