Euler's theorem relate to the remainder of various powers and has applications ranging from modern cryptography to recreational problem-solving. See more

\[\large a^{11762}\equiv{a^2}\pmod{25725}\]

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.

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