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Number Theory

Euler's Theorem

Euler's Theorem: Level 3 Challenges


How many positive integers \(n>1\) evenly divide \(a^{13}-a\) for all positive integers \(a?\)

Find the sum of all prime numbers \(p\) such that \(p|\underset { p }{ \underbrace { 111\dots 1 } } \).

\[\Huge {\color{blue}9}^{{\color{green}8}^{{\color{red}7}^{{\color{brown}6} ^{\color{magenta}5}}}}\]

What are the last two digits when this integer fully expanded out?

Let \(P\) be product of all positive integers less than 720 which are relatively prime to 720. What is the remainder when \(P^2\) is divided by 720?

Find the tens digit of \[ \Large 2014^{2014^{2014}}. \]


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