Number Theory
# Euler's Theorem

$\large a^{11762}\equiv{a^2}\pmod{25725}$

Find the smallest positive integer $a$ such that the congruency above fails to hold.

What is the remainder when

$1^{2016}+2^{2016}+\cdots+2016^{2016}$

is divided by 2016?

Suppose it takes $S$ digits in the binary expansion of $\frac{1}{{3}^{10}}$ for it to repeat. What is the minimum value of $S$?

As an example, $\frac{1}{3} = 0.01010101...$ in base two, which takes two digits to repeat.