I'm having difficulty in understanding a proof given on wikipedia under "Examples" section. The proof can be found at:
Coset - Wikipedia, the free encyclopedia
If is an additive group of integers and a subgroup such that
where is an integer. Then prove that the cosets of in are the sets.
Let be the additive group of integers and the subgroup
where is a positive integer.
Then the cosets of in are the sets , where .
There are no more than cosets, because . <--------------My problem?
The coset is the congruence class of a modulo
Why is that the ? Specifically why's that ?
I know it's true visually but how do I prove that ?
Is it possible to kindly help me find the reason why's and what it got to do with congruence class of modulo which is stated at the next line of the proof above?