Mid '|' in math?

Question

What does this equation mean? What does the $|$ mean?

$446617991732222310 | mn(m^k - n^k)$

Here is the complete question for reference -

What is the smallest positive integer $k$, such that for every ordered pair of integers $(m, n)$, we have

$446617991732222310 | mn(m^k - n^k)$?

Please help

Answer 1

For two integers $a,b$, $$a\mid b$$ means $a$ divides $b$, i.e. $b=ak$ for some integer $k$.

Answer 2

$a|b$ means that $a$ divides $b$.

Or equivalently $b=n.a$ for some $n \in \mathbb{Z}$.

Answer 3

As others have mentioned, the symbol "$\mid$" means division (oddly enough, your question title actually somewhat refers to how "$\mid$" is typeset--this symbol is typeset by using the command $\mid$). More specifically, the notation $x_1\mid x_2$ means "$x_1$ divides $x_2$," where $x_1$ and $x_2$ are integers and we can represent $x_1\mid x_2$ algebraically as $x_2 = \ell\cdot x_1$, where $\ell$ is an arbitrary integer.

In the context of your problem, you are considering $$ 446617991732222310\mid mn(m^k - n^k),\tag{1} $$ where $m,n,k$ are integers. What this means, per the description above, is simply that there exists an integer $\ell$ such that $$ mn(m^k - n^k) = \ell\cdot 446617991732222310. $$

Side note: One typographical thing you should be aware of is that using $\mid$ is different from just using $|$: the spacing is different. Using $\mid$, we get the properly spaced expression in $(1)$. If we just use $|$, then we get the wonky spacing as follows: $$ 446617991732222310 | mn(m^k - n^k). $$ I would edit your question to make this correction, but I'll leave it so you may see the difference.