# Show that ntimes 37−n is divisible by 19 for any integer n.

Show that $n×37-n$ is divisible by 19 for any integer n.

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izboknil3

To show: ${n}^{37}-n$ −n is divisible by 19. We can assume that n is co-prime to 19, because if n is divisible by 19 then we have nothing to prove. We know that

Now we have consider nn, where $gcd\left(n,19\right)=1$, so, by Euler's theorem we cqan say that ${n}^{\varphi \left(19\right)}\equiv 1$ in modulo 19. Now
${n}^{37}-n=\left({\left({n}^{18}\right)}^{2}\right)n-n\equiv {1}^{2}×n-n=n-n=0$
in modulo 19. Hence we are done that is ${n}^{37}-n$ is divisible by 19, for each positive integer n.