Hey all, I`ve been working at this "proof" for several hours now, have put it away several times thinking that maybe I`ll get it if I leave it alone for a bit...has not worked =] It has 2 parts, I think I have proven the first part, but the second one really just stumps me =| 1. Show that a residue class [a] module mn is a subset of the residue class [a] modulo m. This defines a homomorphism Z/mn [tex]\rightarrow[/tex] Z/m, where Z is the set of integers. A similar construction gives a homomorphism from Z/mn to Z/n. Now, use the above maps to construct a homomorphism Z/mn [tex]\rightarrow[/tex] Z/m x Z/n coordinatewise, i.e., from f:R [tex]\rightarrow[/tex] S and g:R[tex]\rightarrow[/tex] T, each x [tex]\in[/tex] R determines (f(x),g(x)) [tex]\in[/tex] S x T. This must shown to be a homomorphism if f and g are. Okay, I believe I have a "proof" of the first part. Here it goes: A residue class [a] mod mn contains all integers b such that mn | (a-b). Now, by definition, m | mn. Since m|mn and mn|(a-b), we see that m|(a-b) (if a|b and b|c then a|c). Hence, all integers that are members of [a] mod mn are members of [a] mod m, so [a] mod mn is a subset of [a] mod m. I believe the same method is used to prove that [a] mod mn is a subset of [a] mod n. QED (i think =)) As for the second part, I don't even know where to begin. Any input/help would be greatly appreciated!