The number of solutions $(P, Q)$ of the equation $\frac{P}{Q}-\frac{Q}{P}=\frac{P+Q}{P Q}$, where $P$ and $Q$ are integers from 1 to 9 inclusive, is
(A) 1
(B) 8
(C) 16
(D) 72
(E) 81
The number of solutions $(P, Q)$ of the equation $\frac{P}{Q}-\frac{Q}{P}=\frac{P+Q}{P Q}$, where $P$ and $Q$ are integers from 1 to 9 inclusive, is
(A) 1
(B) 8
(C) 16
(D) 72
(E) 81

## Solution

If we simplify the rational expression on the left side of the equation and then factor the resulting numerator as a difference of squares we obtain,

$$
\frac{(P-Q)(P+Q)}{P Q}
$$

The equation can now be written as,

$$
\frac{(P-Q)(P+Q)}{P Q}=\frac{P+Q}{P Q} \quad \text { or } \quad P-Q=1 \quad(P Q \neq 0 \text { and } P+Q \neq 0)
$$

The only integers that satisfy this are: $(2,1),(3,2),(4,3), \ldots,(9,8)$.

Thus there are 8 possibilities.

ANSWER: (B)