8. The function $f$ defined on ordered pairs of positive integers satisfies the following three properties:
$f(x, x)=x, f(x, y)=f(y, x)$ and $(x+$ $y) f(x, y)=y f(x, x+y)$. Try to compute $f(14$, 52 ).
8. (364)

By property three, we get $f(x, x+y)=\frac{x+y}{y} f(x, y)$. We will repeatedly use this property, applying property two when the first number of the ordered pair is greater than the second, and finally applying property one, to get
$$
\begin{aligned}
& f(14,52)=f(14,14+38)=\frac{52}{38} f(14,38) \\
= & \frac{52}{38} \cdot \frac{38}{24} f(14,24)=\frac{52}{24} f(14,10) \\
= & \frac{26}{5} f(10,14)=\frac{26}{5} \cdot \frac{14}{4} f(10,4) \\
= & \frac{91}{5} f(4,10)=\frac{91}{5} \cdot \frac{10}{6} f(4,6)=\frac{91}{3} \cdot \frac{6}{2} f(4,2) \\
= & 91 f(2,4)=91 \cdot \frac{4}{2} f(2,2)=364 .
\end{aligned}
$$