Prove that the $12^{t h}$ power of any number is of the form $13 n$ or $13 n + 1$ .

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Solution

if $p$ is a prime number and $N$ is prime to $p$ then $N^{p - 1} - 1$ is a mutiple of $p$
Put $p = 13$
$\Rightarrow N^{13 - 1} - 1 = 13 n \Rightarrow N^{12} - 1 = 13 n \Rightarrow N^{12} = 13 + 1$
Now if $N$ is mutiple of $13$ then obviously
$N^{12} = 13 n$
Hence proved.

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