Strayer Elementary Number Theory 1. 5. 73.

This one got me. 

I knew almost immediately it had to be n!. I knew n! had to be the largest of this kind (for if there were a larger, it would have to also divide n!... how is that possible?). But I didn't know how to prove it. Problem is, I'm still pretty narrow-minded in my mathematics. Outside of analysis, I don't think about derivatives. Outside of algebra, I don't think about groups. And outside of combinatorics, I don't think about factorials. So I often forget them in the context of number theory, and I get bit for it. I know to be so narrow-minded in math is a major shortcoming... I'm working on it :/

So with much thanks to proof wiki, here is the fleshed out solution