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# Number Theory Warmups

If numbers aren't beautiful, we don't know what is. Dive into this fun collection to play with numbers like never before, and start unlocking the connections that are the foundation of Number Theory.

examples, and problems from the community.

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

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Calvin Lin**

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

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Chung Kevin**

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Evaluate

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

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

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Curtis Clement**

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

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Aaditya Bhatt**

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

What are the last three digits of the number above?

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Kalpok Guha**

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