Number Theory

Euler's Theorem

Euler's Theorem: Level 3 Challenges


An army of worker ants was carrying sugar cubes back into their colony. In there, the ants put 1 sugar cube into the first room, 2 into the second, 4 into the third, and doubling the amount so on until the 101th101^\text{th} room.

Then the queen ant decided to build bigger cubic blocks of 5×5×55\times 5\times 5 sugar cubes from all they had previously collected. How many sugar cubes would remain after all these build-ups?

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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