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.
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$\begingroup$ 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. $\endgroup$– Thomas AndrewsNov 14, 2020 at 3:11
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$\begingroup$ 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 $\endgroup$– user71207Nov 14, 2020 at 3:11
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5$\begingroup$ 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.” $\endgroup$– Thomas AndrewsNov 14, 2020 at 3:13
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1$\begingroup$ 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. $\endgroup$– imranfatNov 14, 2020 at 3:14
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$\begingroup$ Yes, is a multiple of is also true. So does it mean "is a multiple of" but not necessarily "is divisible by"? $\endgroup$– user71207Nov 14, 2020 at 3:15
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1 Answer
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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.
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$\begingroup$ thanks! Just curious however, what makes you think it is intuitive? $\endgroup$ Oct 15, 2021 at 9:28
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$\begingroup$ @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. $\endgroup$ Oct 15, 2021 at 15:56
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1$\begingroup$ I had thought that you can say "$\exists z \in \mathbb{Z} : a = zb$"? Not just $\mathbb{N}$ $\endgroup$ Oct 24 at 2:31