# here's why...Thanks to Mr W W W Google-Com

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Posted by Denis Borris on September 07, 2002 at 18:29:08:

In Reply to: I don't know, you guys are throwing but I don't seem to be catching. Let's look at examples... posted by Joel on September 07, 2002 at 15:30:12:

To show that perfect squares have an odd number of factors we express
the number in its prime factors. If it is a perfect square the power
of each prime factor must be even, e.g. 2^2 x 3^4 x 5^2 so that the
square root is given by 2 x 3^2 x 5.

Consider the number 2^2 x 3^4 x 5^2. The number 2 could be chosen
0,1,2 times,i.e. in 3 different ways, the number 3 could be chosen
0,1,2,3,4 times, i.e. in 5 different ways, and similarly the number 5
could be chosen in 3 different ways. So total number of ways that
factors could be made up is given by 3 x 5 x 3 = 45 which is an odd
number. Note that taking none of 2,3 or 5 as factors gives the 1
which we require as a factor. Taking all the numbers 2, 3, 5 to their
highest power gives the number itself - again one of the factors we
require. Thus perfect squares always have an odd number of factors,
and all other numbers have an even number of factors.

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