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

Emeli Hagan 2020-11-29 Answered

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

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Expert Answer

izboknil3
Answered 2020-11-30 Author has 99 answers

To show: n37n −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
ϕ(19)=Z={a: where gcd(a,19)=1,1an}=18.
Now we have consider nn, where gcd(n,19)=1, so, by Euler's theorem we cqan say that nϕ(19)1 in modulo 19. Now
n37n=((n18)2)nn12×nn=nn=0
in modulo 19. Hence we are done that is n37n is divisible by 19, for each positive integer n.

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