How many ordered triples of positive integers satisfy ?

Source: NCTM Mathematics Teacher 2006

**SOLUTION**

This problem is similar to the problem “Triple ABC” posted on 9/14/2012, nevertheless a different solution is offered here.

Given that , the positive integers must take the form , , and where are non-negative integers.

The more difficult problem of finding the single product turns into the much simpler problem of finding two products, the first being such that and the second such that .

If have possible values

If have possible values

If have possible values

If have possible values

If have possible values

If have possible value

Total = possible ways to choose

Similarly, we want .

If have possible values

If have possible values

If have possible values

If have possible value

Total = possible ways to choose

There are ways to choose ; there are ordered triples such that .

**Answer**: