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# Euler's Theorem

Euler's theorem relate to the remainder of various powers and has applications ranging from modern cryptography to recreational problem-solving.

# Euler's Theorem Warmup

Is there a positive integer $$n$$ such that $$2^n \equiv 1 \pmod{7} \, ?$$

$\frac{1}{15}, \frac{2}{15}, \frac{3}{15}, \ldots, \frac{14}{15}, \frac{15}{15}$ How many of these fractions cannot be reduced?

Is 999999 divisible by 7?

Hint: Fermat's Little Theorem states that if $$p$$ is prime and $$a$$ is not a multiple of $$p,$$ then $a^{p-1} \equiv 1 \pmod{p}$

What is the last digit of $$3^{100} \, ?$$

Which of these is congruent to $$10^{100} \pmod{11} \, ?$$

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