Why perfect square has odd number of factors

Question

can someone please describe me why only the perfect square has odd number of factors.why does other number not has odd numbers of factors? I understand it but don't find any mathmetical proof.Please help me

Answer 1

For a given number $n$ we can group its divisors in pairs $(d,\frac nd)$, except that if $n=m^2$ this would pair $m$ with itself.

Answer 2

Well, without the square root, the square no. would be even. Since the square root is multiplied by itself, then there is only 1 more factor, not 2.

Example: 64. Has 6 factors (excluding 8 (squared)). With 8, there is now 7 factors... but not eight because the root (in this case, 8) will be repeated, and there is simply no point in repeating it.

I hope your question is answered! :)