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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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