The Factors of Perfect SquaresDate: 04/26/2008 at 15:20:17 From: Sarah Subject: perfect squares and factors Why are perfect squares the only numbers with odd numbers of factors? Like 16 the factors are 1,2,4,8,16, so yes it has 5 factors. I know that this is true but I don't know why.... please help. Thank you. Date: 04/27/2008 at 16:14:07 From: Doctor Achilles Subject: Re: perfect squares and factors Hi Sarah, Thanks for writing to Dr. Math. Let's ask a related question: why do numbers that aren't squares always have an even number of factors? The answer is that the factors of a number come in pairs. For example, the factors of the number 20 are: 1, 2, 4, 5, 10, and 20. That's because you can write the number 20 as: 1 x 20, 2 x 10, or 4 x 5 For any number that is not a perfect square every factor will have a partner that you multiply it by in order to get the number. Therefore, each factor has a partner and therefore the number of factors is always even. For a perfect square, there will always be a lone factor who doesn't have a partner. The question you have to answer is "why is that?" Hope this helps. If you have other questions or you'd like to talk about this some more, please write back. - Doctor Achilles, The Math Forum http://mathforum.org/dr.math/ Date: 04/27/2008 at 18:39:41 From: Sarah Subject: Thank you (perfect squares and factors) Thanks! This helped a lot on answering my question. Sarah |
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