12.45 The sum, difference, product, and quotient of two integers add up to 450. Find these two numbers.
(Kyiv Mathematical Olympiad, 1946)
[Solution] Let the two integers be $x$ and $y$. Then
$$
x+y+(x-y)+x y+\frac{x}{y}=450 .
$$

Since $x$ and $y$ are integers, $x+y, x-y, x y$ are all integers. Therefore, from (1), $\frac{x}{y}$ is an integer.
Equation (1) can be rewritten as
$$
\begin{array}{c}
\frac{x}{y}+2 x+x y=450, \\
\frac{x}{y}\left(1+2 y+y^{2}\right)=450, \\
\frac{x}{y}(1+y)^{2}=450 .
\end{array}
$$

Since
$$
450=1 \cdot 2 \cdot 3^{2} \cdot 5^{2} .
$$

Then $\frac{x}{y}$ can only be $2, 18, 50, 450$.
Thus, from (2), the values of $x$ and $y$ can be obtained as
$$
\begin{aligned}
(x, y)= & (28,14),(-32,-16),(72,4),(-108,-6), \\
& (100,2),(-200,-4),(-900,-2) .
\end{aligned}
$$

Therefore, there are seven solutions to this problem.