Why perfect square has odd number of factors [closed]

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

Solution 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.

Solution 2:

You can always list the factors of a number, N, into pairs $(a_i,b_i)$ where $a_i \le \sqrt N \le b_i$. This means that a number will always have an even number of factors, unless the number is a perfect square, in which case one pair will consists of the same two numbers. The two examples below should demonstrate why.

\begin{align} \text{factors} &\; \text{of $36 = 6^2$} \\ \hline 1 &,\, 36 \\ 2 &,\, 18 \\ 3 &,\, 12 \\ 4 &,\, 9 \\ 6 &,\, 6 & \text{A total of $9$ factors} \\ \hline \end{align}

\begin{align} \text{factors} &\; \text{of $12$ } \\ \hline 1 &, 12 \\ 2 &, 6 \\ 3 &, 4 & \text{A total of $6$ factors} \\ \hline \end{align}

Solution 3:

I agree that it is counter-intuitive. The catch is that when "Factors" is written, generally what is actually meant is "unique factors". Therefor if a number is a perfect square, it will have an even number of total factors, but odd number of UNIQUE factors, which is what is truly meant.

E.g. 36: 1x36, 2x18, 3x12, 4x9, 6x6 - 10 total numbers but only 9 unique ones. For fun we just decide to ignore the second '6'. Hope that helps.