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Number Theory
# Number Theory Warmups

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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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