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

For prime number $\displaystyle p$ we have :

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

where d is a divisor function .

Quote:

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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

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