The Factors of Perfect Squares

Date: 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