Sum of factors of even positive and perfect square number N

[goelmohit2002]

Hi All,

Can some one please tell what is the "Sum of all the positive factors of a even positive and perfect square number N"...e.g. N = 36, 64, 100

so what is the sum of all the positive factors of N.

is it
a) even
b) odd...

If we see the same in the above examples, it comes to be odd....

But can someone please tell how to prove the same mathematically that it is always odd.

[adam.roffman]

I might be wrong, but I would think the answer would be even. Factoring any perfect square will be, for lack of a better term, symmetrical, right? So if you think of factoring 100 with vertical branches, ie, 10 on one side, 10 on the other, and then 5 and 2 on one side and 5 and 2 on the other (sorry if this is a little convoluted -- hopefully you'll get what I'm trying to say), both the left and right sides will be identical. So whatever factors you have on one side, whether they sum to an odd or an even, will always be doubled to get the sum of all the factors (and hence, the result will always be even).

[Ben Ku]

Please refer to this post, where we were able to show that the sum of all factors of a perfect square is odd.

is-the-positive-integer-n-a-perfect-square-t7551.html?hilit=sum%20factors%20perfect%20square%20odd%20even

Let me know if you have any questions. Thanks!