Number Theory

Number Theory Warmups

Number Theory Warmups: Level 1 Challenges


What is the last digit of \( 2^9?\)

If today is a Monday, then what day will it be 100 days later?


\[\frac{(10!+9!)(8!+7!)(6!+5!)(4!+3!)(2!+1!)}{(10!-9!)(8!-7!)(6!-5!)(4!-3!)(2!-1!)} . \]

Hint: You don't need to multiply out all the factorials!

How many rotations will the gear with 8 teeth have made when the gear with 10 teeth completes its 48th rotation?

\[\LARGE \color{blue}5^{\color{green}5^{\color{red}5^{\color{brown}5^\color{magenta}5}}}\]

What are the last three digits of the number above?


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