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/ |
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