# how does this work? **URL:** \ **Category:** NYSpeed Off Topic\ **Created:** [November 21, 2007, 7:19am UTC](https://forums.speedlife.net/t/how-does-this-work/39077 "2007-11-21T07:19:36Z")\ **Posts on this page:** 1\ **Showing post:** 11
**Author:** ![FightinMike](https://avatars.discourse-cdn.com/v4/letter/f/7bcc69/32.png) [@FightinMike](https://forums.speedlife.net/u/FightinMike)\ **Post date:** [November 23, 2007, 2:36pm UTC](https://forums.speedlife.net/t/how-does-this-work/39077/11 "2007-11-23T14:36:35Z")
This is one of the first things we study in Number Theory. Its a counterintuitive result of residue systems in a congruence class, it works for 2,3 any mutiple of them. For the actual proof you need a few theorems that deal with Complete Residue Systems and Reduced Residue Systems modulo any integer. Modulo 9 the CRS={0,1,2,3,4,5,6,7,8} and the RRS={1,2,4,5,7,8}. Since every integer can be expressed as one of {3m,3m+1,3m+2} where m is some integer, we will always produce a member of the RRS for any integer. There are a few ways to do this next part:1.) we can show through exhausting every case that for any integer modulo 9 we will subtract its congruence when we add its representative integers 2.)We can prove this as a generality for cyclic groups with generators. If you think this is long, the real proof takes a few pages… --- _[View the full topic](https://forums.speedlife.net/t/how-does-this-work/39077)._