# What does $|$ mean?

I came across this notation in a solution manual. It appears in the solution to Exercise 1.2.2. as $$3|2^p$$. Any idea what it could mean?

• Generally it means $3$ divides $2^p$ Commented Sep 6, 2021 at 17:14
• I think you should put that as the answer Commented Sep 6, 2021 at 17:16
• It can also be seen as "such that" in some places too though. But in the context they said, it definitely means "divides" as said. Commented Sep 6, 2021 at 17:23

Generally $$a \mid b$$ means $$a$$ divides $$b$$ where $$a,b\ne 0$$ are integers. Since $$\frac{b}{a}$$ is a rational number it is sometime cumbersome to say that $$\frac{b}{a}$$ is an integer even though it means the same thing. Even we can write in words that " $$b$$ is divisible by $$a$$" but $$a \mid b$$ convention is lot more easy to follow.
It means "divides", i.e. "3 divides $$2^p$$. Formally:
There exists an integer $$k\in\Bbb Z$$ such that $$3\cdot k = 2^p$$.
However, depending over which set or ring you are acting, divisibility is defined the same (in a broader context) but might spell out differently in the details, for example when you are over a ring of polynomials like $$\Bbb Z[x]$$ or over the ring of integers of some algebraic number field like $$\Bbb Z[\sqrt{-1}]$$.
By the way, the symbol for "does not divide" is $$\nmid$$.
• You meant to say $k \cdot 3 = 2^p$. Commented Sep 6, 2021 at 17:25