Finding Perfect Cube Factors of 10!

Date: 04/12/2005 at 11:41:17
From: Tom
Subject: Factorials?

What perfect cubes divide 10!?  Explain your thinking. 

I am confused on how to solve this problem.  I understand that 10! = 
3628800, but I am not sure what cubes would divide this number since 
there are many cubes.

Date: 04/12/2005 at 11:47:01
From: Doctor Ian
Subject: Re: Factorials?

Hi Tom,

Let's look at a simpler example.  How many squares divide 6!?

Rather than evaluating 6!, let's break it into prime factors:

  6! = 6 * 5 * 4 * 3 * 2 * 1

     = (3*2) * 5 * (2*2) * 3 * 2 * 1

     = 5 * 3 * 3 * 2 * 2 * 2 * 2 * 1

We have 3 appearing twice, so 3^2 must divide 6!:

     = 5 * 3 * 3 * 2 * 2 * 2 * 2 * 1         3^2 divides 6!
           =====

We have 2 appearing twice, so 2^2 must divide 6!:

     = 5 * 3 * 3 * 2 * 2 * 2 * 2 * 1         2^2 divides 6!
                   =====

We also have (2*2) appearing twice, so (2*2)^2 must divide 6!:

     = 5 * 3 * 3 * 2 * 2 * 2 * 2 * 1         (2*2)^2 divides 6!
                   =============

Note that 1 is a perfect square, and it divides 6!, even though this
method doesn't identify it.

Is this enough to get started on your problem? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/