can you help me prove this one
Given m > 0. there are exactly m distinct residue classes modulo m, namely,
thank you in advance!
God bless you!
Not sure exactly what you mean. But I try anyway.Originally Posted by earlkaize
Consider the integral domain, , regocnize that it is a commutative ring with unity (by definition) therefore you can speak of ideals. Note that for any , that the coset is an ideal in . Form a factor ring, . And you end with,
Note, all the intergers are divided among these cosets. Further, since cosets are equivalence classes they are all disjoint! Proof complete.
The ring formed by these cosets behaves just like the number under addition modulo , in fact, it is a famous result that you should memorize that,
Well, consider an integer m. Euclidean division grants us that, for all integers n, there are only m cases for the residue when n is divided by m:
n=km, or n=km+1, or n=km+2,..., or n=km+m-1.
The numbers n included in every case form the residue classes, m in total.
This is the core of the idea, as PH explained (though his tough algebra eludes us!)