Number Theory
# Number Theory Warmups

$2^9?$

What is the last digit ofIf today is a Monday, then what day will it be 100 days later?

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!

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

What are the last three digits of the number above?