Show that $a^{x} + a$ and $a^{x} - a$ are always even, whatever $a$ and $x$ may be.

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Solution

If $a$ is odd then $a^{x}$ will also be odd
$\Rightarrow a^{x} + a$ and $a^{x} - a$ will be even as sum and difference of two odd numbers is always even.
If $a$ is even then $a^{x}$ will also be even
$\Rightarrow a^{x} + a$ and $a^{x} - a$ will be even as sum and difference of two even numbers is always even.
Hence proved.

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