Number Theory

Euler's Theorem

Euler's Theorem Warmup

         

Is there a positive integer nn such that 2n1(mod7)?2^n \equiv 1 \pmod{7} \, ?

115,215,315,,1415,1515\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=1061.999999 = 10^6 - 1.

What is the last digit of 3100?3^{100} \, ?

Which of these is congruent to 10100(mod11)?10^{100} \pmod{11} \, ?

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