>>11831839>there are only n possibilities when it comes to remainders, namely 0,1,2,...,n-1, so by gluing together the integers in this way you are left with a 12 element setwoops, n* element set, i must've been thinking of Z/12Z
the most basic example for factor groups are the even and odd numbers. you know from school that there are certain additive (including multiplication would make it a ring) relationships between them, right, namely, even+even=even, even+odd=odd and odd+odd=even. what you've done is you've sorted all the evens and odds into 2 sets, E and O, and now you're naturaly lead to defining addition on these 2 sets, making E+E=E, E+O=O & O+O=O. so by glueing together integers based on their parity you've produced a 2 element group, {E, O}. mathematically speaking you've factored Z by its normal subgroup 2Z (consisting of elements of the form 2n, where n is an integer), producing Z/2Z, 2 element factor group
