Prove that if x and y are both odd positive integers then
Question:

Prove that if $x$ and $y$ are both odd positive integers then $x^{2}+y^{2}$ is even but not divisible by 4 .

 

Solution:

Let, $n$ be any positive odd integer and let $x=n$ and $y=n+2$.

So, $x^{2}+y^{2}=(n)^{2}+(n+2)^{2}$

Or, $x^{2}+y^{2}=n^{2}+\left(n^{2}+4+4 n\right)$

$\Rightarrow x^{2}+y^{2}=2 n^{2}+4+4 n$

$\Rightarrow x^{2}+y^{2}=2\left(n^{2}+2+2 n\right)$

$\Rightarrow x^{2}+y^{2}=2 m$ (where $m=n^{2}+2 n+2$ )

Because $x^{2}+y^{2}$ has 2 as a factor, so the value is an even number.

Also, because it does not have any multiple of 4 as a factor, therefore, it is not divisible by 4.

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