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Is there a special name in number theory for an integer a that is multiple of an integer m, but such that the quotient a/m is no longer divisible by m? Would it be a good idea to call such an m-multiple a SIMPLE (or maybe a PROPER) m-multiple? For example 6 is a 'simple' multiple of 2 because 6/2=3 is not divisible by 2. But 4,8,12, for example, are not 'simple' multiples of 2, because their quotient by 2 is still divisible by 2. I am currently interested in the orbits under the action of modulo-n multiplication of the multiplicative group of n-totatives on the set of all n-residues. There is an orbit for each n-divisor m < n. It consists exactly of the positive 'simple' multiples of m less than n, because any m-multiple that is not 'simple' is already in the orbit of a multiple of m. The set of 'simple' m-multiples less than n (for an n-divisor m) has also several other interesting properties in connection with the set of n-totatives. So I would like to have an appropriate name for it.

frank
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    I would say that $a$ is $m^2$-free. – TheSilverDoe Aug 26 '20 at 16:04
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    I know that the symbol for it is $m||a$ – Exodd Aug 26 '20 at 16:22
  • In the case that $m$ is a prime number , we can also say that the $m$-valuation of $a$ is $1$. But for composite $m$, I think, there is no such name. "$m^2-free$" is a good idea. – Peter Aug 26 '20 at 16:23
  • "proper multiple of m" or "proper m-multiple" would rather be any multiple of $m$ apart from $m$ itself. – Peter Aug 26 '20 at 16:27
  • I would say, "the highest power of $m$ that divides $a$ is $1$". I believe there is a term for "the highest power of a number that divides another" but I don't remember it. – fleablood Aug 26 '20 at 16:29
  • @fleablood the highest power of $x$ that divides $y$ is the $x$-adic valuation of $y$. – Rhys Hughes Aug 26 '20 at 16:40
  • @RhysHughes yeah.... that's easy to remember and when you use it everyone no matter how novice will know what you mean.... But, yes, that was the term. So $6$ has 2-adic valuation of $1$ is valid terminology. ... I thought there was another. I occasionally use "order" but I believe that's a slight abuse.... Actually I that there was a number that refered to the $2$ in terms of the $6$. $2$ aditic valuation refers to the $6$ in terms of the $2$. – fleablood Aug 26 '20 at 17:14
  • @fleablood I thought "valuation" would only apply for prime numbers. But it would make sense to generalize this terminology. – Peter Aug 26 '20 at 17:36

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To summarise, you want an expression of the relationship: $$m|a \text{ and } m^2\nmid a$$ We would say that the $m$-adic order of $a$ is $1$, that is that $m=m^1$ is the greatest power of $m$ that divides $a$. This is written as $\nu_m(a)=1$.

J. W. Tanner
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Rhys Hughes
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