Competition problem — Number theory

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Number theory Difficulty 5.3 AIME, harder Find the answer

5. Find all solutions to the congruence equation $5^{x} \equiv x(\bmod 23)$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

5. $x \equiv a(\bmod 22), x \equiv 5^{a}(\bmod 23), 0 \leqslant a \leqslant 21$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.