Competition problem — Number theory

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Number theory Difficulty 5.4 AIME, harder Find the answer

26. Find the least positive residue of
a) $3^{10}$ modulo 11
b) $2^{12}$ modulo 13
c) $5^{16}$ modulo 17
d) $3^{22}$ modulo 23 .
e) Can you propose a theorem from the above congruences?

Your answer

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

Solutions — 2

Solution 1

26. a) 1 b) 1 c) 1 d) 1 e) $a^{p-1} \equiv 1(\bmod p)$ when $p$ is prime and $p \nmid a$

Solution 2

26. a) 1 b) 1 c) 1 d) 1 e) $a^{p-1} \equiv 1 \pmod{p}$ when $p$ is prime and $p \nmid a$

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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.