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Is there a standard way of writing $a$ is divisible by $b$ in mathematical notation?

From what I've search it seems that writing $a \equiv 0 \pmod b$ is one way? But also you can write $b \mid a$ as well (the middle character is a pipe)? And sometimes that pipe is replaced by $3$ vertical dots?

Or is there a way of writing $a$ is a multiple of $b$ which I think means the same thing?

EDIT: thanks for the answers, is there a way to extend this and write something like: $b \mid a$ when $a = k$

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  • $\begingroup$ I usually use $|$ $\endgroup$
    – Belgi
    Apr 22, 2012 at 12:48
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    $\begingroup$ The standard is $b\mid a$ whic is typeset "b \mid a" in TeX $\endgroup$
    – Andrea Mori
    Apr 22, 2012 at 12:54
  • $\begingroup$ en.wikipedia.org/wiki/List_of_mathematical_symbols $\endgroup$
    – Pedja
    Apr 22, 2012 at 12:56
  • $\begingroup$ @pedja, thanks but that article lists both options $\endgroup$
    – Jonathan.
    Apr 22, 2012 at 13:03
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    $\begingroup$ @PeterPhipps The spacing is different, compare $A|B$, $A\mid B$, $A\vert B$, $A\lvert B$ and $A\rvert B$ (edit: there actually are differences there in some fonts, even if those are not visible here). $\endgroup$
    – dtldarek
    Apr 22, 2012 at 13:17

7 Answers 7

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I have seen the following:

  • $b \mid a$ that is with $\LaTeX$ \mid
  • $a = 0 \mod b$ that is with $\LaTeX$ \mod
  • $a = 0 \pmod b$ that is with $\LaTeX$ \pmod
  • $a \bmod b = 0$ that is with $\LaTeX$ \bmod
  • $a \equiv 0\ (b)$
  • $a \equiv_{b} 0$

and of course there is

  • $a = bk$ for some $k \in \mathbb{Z}$

Choose whatever suits you (and your friends or readers) best!

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    $\begingroup$ Thanks for the list, a mod b = 0 makes the most sense to me. But b|a seems more for use in commentary? $\endgroup$
    – Jonathan.
    Apr 22, 2012 at 13:19
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    $\begingroup$ It depends on so many things that I can't tell you this or that way. For me there are three important factors: how often will I use it (more often means less symbols), do I need to use it "in chains" like $a = b = c = d \pmod n$ and do I need to use different $n$-s, e.g. $a \equiv_3 b \equiv_5 c \equiv_7 d$ (which may be confusing but sometimes is helpful). Still, the most important criterion of all is readability. $\endgroup$
    – dtldarek
    Apr 22, 2012 at 13:26
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Alexander Merkurjev taught me a long time ago the ingenious Russian notation $6 \vdots 2$, which I immediately adopted .
It pleasantly "rhymes" with the equivalent $(6)\subset (2)$

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    $\begingroup$ I wonder if you are the unique non-Russian who uses that notation. I didn't think it is used in public outside of Russia (or at least Eastern Europe). $\endgroup$
    – KCd
    Jan 22, 2013 at 13:48
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    $\begingroup$ We use it in Romania too! $\endgroup$
    – Victor
    Mar 24, 2016 at 21:12
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    $\begingroup$ @Victor I recently attended a lecture in the US aimed at undergraduates and the lecturer used this notation without being aware that it is practically unknown here. The lecturer was Romanian. $\endgroup$
    – KCd
    Oct 17, 2021 at 4:27
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There is also " $a \in b\mathbb Z$ ".

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I often write that as b divides a

Notation:

$$b \mid a$$

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    $\begingroup$ This is the standard way, in the specific meaning of compliance to international standards: ISO 80000-2, clause 2.7-17. Note that the vertical bar character used there is normatively identified as U+2223 DIVIDES (∣), note the common U+007C VERTICAL LINE (|) that we enter directly on a keyboard. It is of course possible to express the same thing using a congruence notation, but only for integers (not e.g. for polynomials). $\endgroup$
    – Jukka K. Korpela
    Jun 22, 2012 at 5:09
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  • Definition: Integer $n$ is a divisible by an integer $d$, when $\exists k \in \mathbb{Z}, n=d\times k$.

  • Notation: $d \mid n$

  • Synonymous:

  • $n$ is a multiple of $d$.

  • $d$ is a factor of $n$

  • $d$ is a divisor of $n$

  • $d$ divides $n$

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$a \equiv 0 \mod b$ and $b \mid a$ are both common, and their use depends on the context. Given a choice, I use the latter more than the former.

There are others such as $\text{lcm}(a,b)=a$ or $\text{hcf}(a,b)=b$ [or perhaps $\text{gcd}(a,b)=b$ if you prefer] which might also be used when more suitable for the context.

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COMMENT.- A notation forget in this page is $a\mid\mid b$ which is pertinent many times, very useful really, in number theory. It is used when a power of a prime, say $p^n$ is the maximal power that divides an integer $M$. In other words $$p^n\mid\mid M\iff p^n\mid M \text{ but } p^{n+1}\nmid M$$

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