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Fermat's little theorem states that if $p$ is a prime number, then for any integer $a$, the number $a^{p}-a$ is an integer multiple of $p$. In the notation of modular arithmetic, this is expressed as $$ a^{p} \equiv a \quad(\bmod p) $$

Using Little Fermat Theorem: find all solutions: $x^{14} \equiv 9(\bmod 13)$

I'm having first experience with non linear congruences so i would be glad if somebody explained my how to make progress with that.

user3231387
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    Can you remind us and yourself what Fermat's Little Theorem says? – JMoravitz Apr 23 '21 at 12:34
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    So, $x^{13}\equiv x\pmod{13}$, yes? So rather than looking for solutions to $x^{14}\equiv 9\pmod{13}$ we can instead look for solutions to $x^{13}\cdot x\equiv 9\pmod{13}$ which by FLT is $x\cdot x\equiv 9\pmod{13}$. Yes? Can you solve this other simpler problem? – JMoravitz Apr 23 '21 at 12:39
  • Substituting $,x^{13}\equiv x,$ yields $,0\equiv x^2-3^2\equiv (x-3)(x+3),$ and you solve that as in the dupe. – Bill Dubuque Apr 23 '21 at 16:03

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