Number Theory

Number Theory Warmups

Number Theory Warmups: Level 1 Challenges


What is the last digit of 29? 2^9?

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


(10!+9!)(8!+7!)(6!+5!)(4!+3!)(2!+1!)(10!9!)(8!7!)(6!5!)(4!3!)(2!1!).\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?

55555\LARGE \color{#3D99F6}5^{\color{#20A900}5^{\color{#D61F06}5^{\color{#624F41}5^\color{magenta}5}}}

What are the last three digits of the number above?


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