Small but worth knowing funda

Why is sum of the factors of an perfect square is always odd?

If you are still guessing then read on

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