I have a set of residueclasses mod $m$, e.g. with $m=11$ one could have the set $M=\left\{[2]_{11}, [5]_{11}, [6]_{11}, [10]_{11}\right\}$. Now I define a ratio $\operatorname{d} :=\frac{\#M}{m} = \frac{4}{11}$.
So in this set I find all the number $2,5,6,10,13,16,17,21,24,27,28,32\dots$ and infinitely more. Now I want to formalize the following: Let $a$ be a natural., e.g. $a=3$. Then instead of $M$ I could use $M'=\left\{[2]_{11}, [13]_{11}, [24]_{11}, [5]_{11}, [16]_{11}, [27]_{11}, [6]_{11}[17]_{11}, [28]_{11},[10]_{11}, [21]_{11}, [32]_{11}\right\}.$ But the radio $\operatorname{d'}=\frac{\#M'}{3\cdot 11}$ would remain the same.
Can anyone provide me with an approach to formalizing this statement?