Number Theory

Euler's Theorem

Euler's Theorem: Level 3 Challenges

           

Several integers are given (some of them may be equal) whose sum is equal to 12801280. Decide whether the sum of their seventh powers can equal 20182018.

How many positive integers n>1n>1 evenly divide a13aa^{13}-a for all positive integers a?a?

Find the sum of all prime numbers pp such that p1111pp|\underset { p }{ \underbrace { 111\dots 1 } } .

98765\Huge {\color{#3D99F6}9}^{{\color{#20A900}8}^{{\color{#D61F06}7}^{{\color{#624F41}6} ^{\color{magenta}5}}}}

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

Let PP be product of all positive integers less than 720 which are relatively prime to 720. What is the remainder when P2P^2 is divided by 720?

Find the tens digit of 201420142014. \Large 2014^{2014^{2014}}.

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