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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

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    $\begingroup$ Division. $a \ | \ b$ means $a$ divides $b$ for two integers, $a, b$ $\endgroup$
    – Ishfaaq
    Commented Mar 26, 2015 at 11:29
  • $\begingroup$ It leaves no remainder? $\endgroup$
    – Confuse
    Commented Mar 26, 2015 at 11:32
  • $\begingroup$ Yes, if you generally draw that distinction. $\endgroup$
    – Ishfaaq
    Commented Mar 26, 2015 at 11:33
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    $\begingroup$ I see in some of the comments and answers that the symbol means 'division'. I think that is a bit of a misstatement. Division is an operation that returns a numerical value (or values, depending on how one handles the remainder). The symbol $\mid$ is for 'divides', which is a relation and gives a value of 'true' or 'false' as in $2 \mid 6$ is true; and $2 \mid 5$ is false. This may seem like a fussy distinction, but perhaps not so much when one sees statements like $2 \mid 6 = 3$. $\endgroup$
    – paw88789
    Commented Mar 26, 2015 at 12:04
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    $\begingroup$ It's one of the most unfortunate notations in all of mathematics: A completely symmetric symbol denoting a highly asymmetric relation. $\endgroup$ Commented Mar 26, 2015 at 12:10

3 Answers 3

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

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

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

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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.

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