Use that $a^5\equiv a\pmod 5$ and if $(a,5) = 1$ then prove $a^4\equiv 1\pmod 5$

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Use that $a^5\equiv a\pmod 5$ and if $(a,5) = 1$ then prove $a^4\equiv 1\pmod 5$

I´ve been trying to solve this but I don’t get any where

Welcome to Math.SE! Please use MathJax. For some basic information about writing math at this site, see MathJax tutorial and quick reference and equation editing how-to. Please read this post for writing a good question. Do you know Fermat's Little Theorem? — GNUSupporter 8964民主女神地下教會, Nov 28 '20 at 22:01
Divide both sides by $a$. You are allowed to do this because $a$ is invertible in $\mathbb{Z}/5\mathbb{Z}$. — Pranav Chinmay, Nov 28 '20 at 22:03
Use the theorem: If $d\mid bc$ and $(d,b)=1$ then $d\mid c.$ Where $a=5, b=a,$ and $c=???$ — Thomas Andrews, Nov 28 '20 at 22:26
If nothing else helps, just calculate $(\pm1)^4$ and $(\pm2)^4$, also, observing $(\pm2)^2\equiv -1\pmod5$ might help. — Berci, Nov 28 '20 at 22:31

score 0 · Answer 1 · answered Nov 28 '20 at 22:54

$a^{5} \equiv a \pmod 5 \implies 5|(a^{5}-a)=a(a^{4}-1)\implies a(a^{4}-1)=5k%$ for some integer $k$. Since $\gcd(a,5) =1,$ and by the uniqueness of prime factorizations, $5$ is a prime factor of the LHS, but not a prime factor of $a$. Hence, $5$ must be a prime factor of $a^{4} -1$ so that $5|(a^{4}-1) \implies a^{4} \equiv 1 \pmod 5.$

Use that $a^5\equiv a\pmod 5$ and if $(a,5) = 1$ then prove $a^4\equiv 1\pmod 5$

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