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(I bet you will send me to english.stackexchange.com and they will send me back here)

I will use an example - I have two lists of integers:

  • list A: 0, 4, 8, 12, ...
  • list B: 2, 6, 10, 14, ...

I'm looking for a neat word to clearly differentiate these lists when talking to a colleague who doesn't really have a maths mindset. My intuition is:

  • list A is "even multiples of 2"
  • list B is "odd multiples of 2"

I tried to google this sort of wording, but I get mixed results:

  • Some results use these words the way I came up with, which is promising;
  • Some results would only call n and odd multiple of k if n itself is an odd number (and thus k is odd too);
  • Some results also use the word "odd" as another word for "unusual" (which is odd, haha)

So what is the term for multiples made by using an odd/even factor?

Hopefully by now you can see why I struggled to come up with a good title for this question.

EDIT: A comment from GerryMyerson hints that I probably used an unfortunate k in my example. I'm after a word that will work for any k, whether or odd or even (although "pateksan's odd multiples" of odd k are actually odd numbers). So for example list A might be 0, 8, 16, 24, ... and list B might be 4, 12, 20, 28, ...

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    $\begingroup$ The first set is $4\mathbb{Z}$ and the next one is $4 \mathbb{Z}+2.$ Or replace $\mathbb{Z}$ with $\mathbb{N}_0$ if you only want non-negative integers. $\endgroup$
    – K.defaoite
    Jul 24, 2020 at 2:35
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    $\begingroup$ Perhaps "non negative numbers that are multiples of four" and "non negative numbers that leave a remainder of two on division by four"? $\endgroup$
    – K.defaoite
    Jul 24, 2020 at 2:40
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    $\begingroup$ $A$ is doubly even numbers, $B$ is singly even numbers. Sez en.wikipedia.org/wiki/Singly_and_doubly_even $\endgroup$ Jul 24, 2020 at 2:43
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    $\begingroup$ @Don I won't disagree with you, but OP asked "what is the term...?" and insofar as there is a term this seems to be it. $\endgroup$ Jul 24, 2020 at 2:52
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    $\begingroup$ If that's what you want, pateksan, then that's what you should ask for. We're mathematicians here, not mindreaders. $\endgroup$ Jul 24, 2020 at 2:53

1 Answer 1

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We use the term modulo to describe what is left when you divide integers by a number $m$. Your example of $0,4,8,12,\ldots$ are the numbers equivalent to $0$ modulo $4$, often written $n \equiv 0 \pmod 4$. The numbers $2,6,10,14,\ldots$ are equivalent to $2 \pmod 4$. Does that do what you want?

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