The user-friendly version of this content is available here.

The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.

This problem trains for: AMC-8, AMC-10, AMC-12.

A number has 120 divisors. The product of all these divisors is:

If a number N has the prime factorization:

equation

Then the number of its positive divisors is:

equation

Let us write explicitly the set of all the divisors as:

equation

The product of all the divisors is:

equation

If the number N is divisible by d, then it is also divisible by:

equation

Therefore, the product of all the divisors can also be written as:

equation

By multiplying the two expressions for the product:

equation

We obtain a general formula for the product of all the divisors of a number:

equation

We can substitute the given number of divisors in this formula to find that:

equation