Number Theory

Euler's Theorem

Euler's Theorem: Level 3 Challenges


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 that divides 1111111111111111the digit 1 is repeated p times\underset{\text{the digit 1 is repeated }p \text{ times}}{\underbrace{111111111111111\cdots 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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