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Let $L$ be a positive integer and consider the integers modulo $L$, denoted $\mathbb{Z}_{L}$. Let $0<n<L$ be a positive integer. I want to think about shifts of the elements of $\mathbb{Z}_{L}$ by $n$, i.e., for $k\in\mathbb{Z}_{L}$, consider the set $G(k)\equiv\left\{k,k+n,k+2n,\cdots\right\}$, where addition is performed modulo $L$.

Now, for each $k$, there exists a set $G(k)$. I want to count the number of such sets (up to permutations) as a function of $L$ and $n$. This might be clarified by some examples:

$L=16, n=6$: In this case, we have $$G(1) = \left\{1,7,13,3,9,15,5,11\right\}$$ $$G(2) = \left\{2,8,14,4,10,16,6,12\right\}$$

Since $G(1)\cup G(2)=\mathbb{Z}_{L}$, there are two sets in this case.

Beyond individual cases like this, there are two general situations where I know the answer to my question. If $L/n=p\in\mathbb{Z}$, then it is clear that there are $p$ sets of interest. Also, if $L$ and $n$ are coprime, then it is clear that there is a single set.

I'm sure questions like this can be better formulated in more abstract-algebraic language, but I'm a hobbyist and don't know more specialized terminology. Thanks in advance for your help.

miggle
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    Do you know any algebra, e.g. cyclic groups or cosets? If not then you may find helpful the intuitive graphic viewpoint here in terms of star polygons (spirograph curves). Your example is the star polygon $,{16/6},$ for which the same argument I gave for $,{10/6}$ applies, since $,\gcd(16,6)=2=\gcd(10,6).,$ These curves make the cyclic structure easily comprehensible even to bright elementary school students. $\ \ $ – Bill Dubuque Jan 24 '25 at 19:39

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