a ∣ b and a ∤ b in formulas

Usually \mid is used for the "is a divisor of" relation. It's just a vertical bar, but declared a relation symbol.

For the negation, \nmid from amssymb is better than \not\mid that doesn't give a good result.

It is surely a good defining semantically sound names, for instance

\usepackage{amssymb}
\newcommand{\divides}{\mid}
\newcommand{\notdivides}{\nmid}