**Triple ABC**

How many ordered triples of positive integers satisfy ?

Source: mathcontest.olemiss.edu 9/10/2012

**SOLUTION**

This problem is similar to the problem “XYZ=4000” posted on 11/5/2014, nevertheless a different solution is offered here.

Let be a positive integer and its prime factorization, then the number of divisors of equals . For example,

Number of divisors

Divisors

Number of divisors

Divisors

We multiply the divisors to find the triples and use multiplication tables to show the products.

multiplication table

The diagonal gives the triples; the diagonal gives the triples; the diagonal gives the triples, and so on. Beware that there will be duplicates. For simplicity, we list the triples in increasing numerical order.

yields distinct triples

multiplication table

yields distinct triples

yields duplicate triples

yields duplicate triples

yields no solution

yields no solution

yields no solution

yields no solution

yields no solution

multiplication table

yields distinct triples

The complete list of the distinct triples is shown below

The problem asks for ordered triples.

give

ordered triples.

The remaining triples give

ordered triples.

Total number of ordered triples such that

**Answer**: