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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 $a = k$

  • 0
    d|A means d divides A.2017-06-12

6 Answers 6

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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!

  • 1
    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.2012-04-22
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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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    We use it in Romania too!2016-03-24
9

There is also " $a \in b\mathbb Z$ ".

6

I often write that as b divides a

Notation:

$b \mid a$

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    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).2012-06-22
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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.