0

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?

Lereu
  • 424
  • Is what you seek a proof that the congruence class $,[i]m\in\Bbb Z_m$ expands to $,a,$ classes $[i]{am},, [i+m]{am},,\ldots,,[i+(a!-!1)m]{am}\in\Bbb Z_{am}?,$ Clearly the ratio claim follows immediately from that. – Bill Dubuque Mar 17 '24 at 09:16
  • @BillDubuque This seems to be what I am looking for. Do you have a proof for me which i can cite in my thesis? – Lereu Mar 17 '24 at 09:22

0 Answers0