Does $x$ include $1$ and $y$ when we say $x\mid y$?

Question

I'm new to number theory. If we have a statement saying $n\mid91$, does it mean that $n$ can be $1, 7, 13, 91$, or n can be $7, 13$. Is $1$ and $91$ excluded?

Answer 1

The bare statement $n\mid91$ does include the possibility that $n=1$ or $n=91$, since it is true that $1\mid91$ and $91\mid91$. Only if we said "$n$ is a non-trivial divisor of $91$" would $n=1,91$ be excluded from consideration.

Answer 2

"$x\mid y$" is shorthand for (or as we mathematicians say it, is defined as) "There is an integer $n$ such that $nx = y$". Nothing about $x\neq 1$ or $x\neq y$ in there. In fact, $1\mid y$ is always true, for any integer $y$, and I can imagine there are contest problems where this is used to show that some unknown number is indeed $1$.

This also means that we get $0\mid 0$, which isn't encountered often, but it's a fun little consequence that one might not consider.