Sum of factors of even positive and perfect square number N

goelmohit2002 wrote: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.

Thanks
Mohit

If Perfect Square N is odd: An odd number will not have even factors and since square number will have odd number of factors. Sum of all the odd factors will be odd

If Perfect Square N is even: This is bit tricky! We have to find number of even and odd factors. Lets say 100. 100=22*52.

To find number of odd factors in 100 take 22 out of 100 (since odd factors will not have even multiple) we are left with 52 which will have 3 factors (2+1). Hence we have 3 odd factors in 100. Number of factors in 100 is (2+1)*(2+1) = 9 (I am adding one to the exponent of 2 and 5 and this is the standard method of finding number of factors)

We found that total number of factors is 9 and there are 3 odd factors hence we will have 6 even factors (9-3). Now 6 even factors added together will be an even number and 3 odd factors added together will be an odd number Hence Even+Odd = Odd

Hope this is clear

Thanks,
Quant-Master

Awesome QM....this is real awesome !!!!!

Thanks a lot !!!

Really helpful.

goelmohit2002 wrote:Awesome QM....this is real awesome !!!!!

Thanks a lot !!!

Really helpful.

Thanks...added this concept to my blog now

Quant-Master