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Implication

Prove that ev...

Question

Prove that every even power of every odd number is of the form $8 r + 1$ .

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Solution

Every odd number is of from $2 x + 1$
${(2 x + 1)}^{2 n} = {{(2 x + 1)}^{2}}^{n} = {(4 x^{2} + 1 + 4 x)}^{2} = {4 x (x + 1) + 1}^{2}$
Now $x (x + 1)$ is always even which will be of form $2 r$
$\Rightarrow {(2 x + 1)}^{2 n} = {4 (2 r) + 1}^{2} \Rightarrow {(2 x + 1)}^{2 n} = {(8 r + 1)}^{2}$
Hence proved.

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