# why is a perfect square has odd number of distinct factors?

• May 8th 2012, 09:59 PM
cmath
why is a perfect square has odd number of distinct factors?
Can some one show me how to proof this? or this is just common sense?

Eg: 36:2,3,6
4: 2
9: 3

Thanks
• May 8th 2012, 10:29 PM
princeps
Re: why is a perfect square has odd number of distinct factors?
For prime number $p$ we have :

$d\left(p^n\right)=n+1$

where d is a divisor function .
• May 9th 2012, 03:16 AM
Sylvia104
Re: Why does a perfect square have odd number of distinct factors?
Quote:

Originally Posted by cmath
Can some one show me how to proof this? or this is just common sense?

Eg: 36:2,3,6
4: 2
9: 3

Thanks

Think of it this way. If $d$ is a divisor of $n,$ then so is $\frac nd.$ The integer $n$ is a perfect square if and only if it has a divisor $d$ such that $d=\frac nd.$