1. The problem statement, all variables and given/known data For the set ℤ, define ~ as a ~ b whenever a-b is divisible by 12. You may assume that ~ is an equivalence relation and may also assume that addition and multiplication of equivalence classes is well defined where e define [a]+[ b ] = [a+b] and [a]*[ b ] = [ab] for all [a],[ b ]. Find a positive integer d such that [d]+= find a positive integer t such that [t]+ = 2. Relevant equations 3. The attempt at a solution These problems seem like a lot of fun. However, I'm not quite getting it. I feel like once I understand one of these, i'll be able to understand all of the easy ones like this. We define a ~b whenever a-b is divisible by 12. So we are saying in the first problem d-5 has to be divisible by 12? If d = 17 then we have 17-5 which is 12 and that is divisible by 12. But how would += In fact, how would any positive integer satisfy that? Since we have well defined addition as [a]+ = [a+b] this would mean [d]+ = [d+5] and this means [d+5] = , but since s must be a positive integer this could not happen... I feel like there must be something clear here that I'm missing and once I get it it will be an easily solvable problem.