diasenamorn5l

2022-04-27

Way to calculate exponent in congruent equation

I want to solve

${5}^{x}\equiv 21\left(\text{mod}23\right)$

Is there a way to get the$x$ without trial & error?

I want to solve

Is there a way to get the

Friegordigh7r7

Beginner2022-04-28Added 16 answers

We have ${5}^{11}\equiv -1\left(\text{mod}23\right)$, because $\left(\frac{5}{23}\right)=-1$, and also ${5}^{2}\equiv 2\left(\text{mod}23\right)$ hence

${5}^{13}\equiv {5}^{11}{5}^{2}\equiv -2\equiv 21\left(\text{mod}23\right)$

And the number $13$ is the smallest because the function $5}^{x$ is periodic modulo $23$ with period $\varphi \left(23\right)=22.$

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