# Does ⋮ mean "is divisible by" in mathematical notation?

For example, $$5(2+3m)⋮5$$ if $$m$$ is an integer. My teacher said yes, but I can't find anything about this triple colon ⋮ online.

• It’s certainly not a common usage. But notation is not universal, and if your professor defines it, that’s what it means in the class, but only in the class. Nov 14, 2020 at 3:11
• ok hmm. What is the usual way of writing it then? It may be awkward if in an exam the marker doesn't know what it means Nov 14, 2020 at 3:11
• The only way in normal notation to write it is $5\mid 5(2+3m).$ I don’t know of a standard symbol for “a is divisible by b.” Nov 14, 2020 at 3:13
• This is also new to me. Maybe he meant to say "is a multiple of..."? Either way, I have not seen the triple colon notation. Nov 14, 2020 at 3:14
• Yes, is a multiple of is also true. So does it mean "is a multiple of" but not necessarily "is divisible by"? Nov 14, 2020 at 3:15

A brief examination of Belarusian, Russian, and Ukrainian Wikipedia pages corresponding to https://en.wikipedia.org/wiki/Divisor suggests that the triple colon "⋮" is a widespread notation meaning "is divisible by" or "is a multiple of" (which are the same). That is,

• "$$a⋮b$$",
• "$$b\mid a$$",
• "$$\exists n\in\mathbb{Z}:a=nb$$",
• "$$a \mod b = 0$$", and
• "$$a = 0\,\pmod b$$"

are all equivalent.

In my opinion, the triple colon notation makes sense and is intuitive, just like the "French brackets".

Your teacher probably comes from the former Soviet Union.

• thanks! Just curious however, what makes you think it is intuitive? Oct 15, 2021 at 9:28
• @user71207 I look at $\div$ and see the horizontal bar dividing something into two parts. I look at " ⋮ " and see that something is divided into three equal parts. Also, of all the expressions I listed, $b \mid a$ is the only one in which $b$ precedes $a$, which might be confusing. Notation " ⋮ " allows you to write the dividend before the divisor, as we normally do. Oct 15, 2021 at 15:56
• I had thought that you can say "$\exists z \in \mathbb{Z} : a = zb$"? Not just $\mathbb{N}$ Oct 24 at 2:31