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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.

# Euler's Theorem: Level 4 Challenges

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

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?

A degree $$5$$ monic polynomial $$P(x)$$ has $$5$$ (not necessarily distinct) integer roots. Given that $$P(0)=2014$$, how many ordered quintuplets of roots $$(r_1,r_2,r_3,r_4,r_5)$$ satisfy the property that $r_1+r_2+r_3+r_4+r_5$ has the same units digit as $r_1^5+r_2^5+r_3^5+r_4^5+r_5^5$

Let $$\phi(n)$$ denote Euler's Totient Function. If the greatest common divisor of the positive integers $$m$$ and $$n$$ is 7, and $$\phi(mn) = 5544,$$ find the least possible value of $$\left|\phi(m)-\phi(n)\right|$$.

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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