How many pairs of positive integers , with and both less than , satisfy the equation ?

Source: NCTM Mathematics Teacher

**SOLUTION
** implies that or .

Suppose

Not possible because is a positive integer.

We conclude that . Since is a positive integer, there exists some positive integer such that .

Squaring both sides of the equation

Divide both sides by

is an integer implies that is an integer. This in turn requires that divides which only occurs when .

Verification

The answers are pairs

**Answer**: 39

**Alternative solution
**

Squaring both sides

Rearrange the terms

Factor both sides

Divide both sides by