# What does the | symbol mean in $3 | n^3 + 5n + 6$?

Using induction prove that:

$$3 | n^3 + 5n + 6$$

I don't understand what the "|" symbol means here - the only thing I could find with google was "given", in probability, but I don't think that applies here.

Usually $n\mid m$ means that $n$ divides $m$ (or $m$ is a multiple of $n$). For example $3\mid 6$ but $3\not\mid 7$.

Writing $a | b$ means "$a$ divides $b$" or "$b$ is a multiple of $a$" or "$a$ is a divisor of $b$", which are all different ways of saying the same thing.

It's read as "divides"; it means that the number on the left divides evenly into the number on the right. More formally, given integers $n$ and $m$, $m\vert n$ provided that there exists $k$ such that $n=km$. For example, $3\,\vert \,6$ since $6=3\cdot 2$. It's not the case that $2\,\vert\, 7$, since no multiple of $2$ is equal to $7$.

• One should read there exists an integer $k$ such that $n=mk$, otherwise the whole definition is meaningless. – Mr. Xcoder Jun 6 '17 at 14:29
• Not completely meaningless... it would mean that $m \neq 0 \lor n = 0$ if I'm not mistaken. But yes, not quite as useful as its actual meaning. – Jasmijn Jun 6 '17 at 14:52

The sign $|$ reads as "divides" and for $x\:|\:y$, it means that $x\in D_y$, or that $y\in M_x$.

$x\:|\:y\implies y\:\vdots\:x\implies \frac{y}{z}\in\mathbb{Z}$

In your question, you have to show that $3$ divides $n^3+5n+6$.