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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.

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

**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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