Number Theory

Euler's Theorem

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: \(999999 = 10^6 - 1.\)

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

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

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