Question

Find the remainder when 1! + 2! + 3! ... + 100! is divided by 30.

Answer 1

First step: Figure out what the least \(n\) is such that \(30\) divides \(n!\). Can you tell me what this \(n\) is?

Answer 2

That number will be greater than the number of molecules in the universe!

Answer 3

5?

Answer 4

Bingo. So we can rewrite is as\[1!+2!+3!+4!+30(\text{huge number})\]So the question is as simple as finding the remainder when you divide \(1!+2!+3!+4!\) by 30.

Answer 5

ohhh the remainder! oops. my bad

Answer 6

Ooh...that is really easy. so the answer would be 3.

Answer 7

Right. Oftentimes problems that look really difficult actually just require a little trick.

Answer 8

Thanks!

Answer 9

You're welcome.