Number Theory

Euler's Theorem

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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