1. The problem statement, all variables and given/known data Let a,b, be positive integers, and let d=gcd(a,b) and m=lcm(a,b). Show ZaXZb isomorphic to ZdXZm 2. Relevant equations m=lcm(a,b) implies a|m, b|m and if a,b|c then m|c. d=gcd(a,b) implies d|a, d|b and if c|a and c|b then d|c let n be an integer with prime decomposition n=p1a1...pkak then Zn=Zp1a1X...XZp1ak 3. The attempt at a solution It is clear that d|m I considered the prime decomposition of a and b as sets, denote as A and B respectively, and consider the intersection, call this set P. intersection(A,B) = P Call the product of the elements in this set P'. I claim P' = gcd(a,b), this should be clear. Then I considered the lcm as the product of the elements in A-P and B-P call these sets A' and B' respectively. Now we can write the d = lcm(a,b) =A'*B'*P' and this should be clear. So... now I want to use the isomorphism as described as in the section above, but I don't know what to do. I think I should use ZaXZb is isomorphic to something like (*) Zp1k1X...XZpikiXZq1l1X...XZpnln. where p are the primes of a and q are the primes of b. Great and now I want to show that that ZdXZm is isomorphic to the same thing, but I'm starting to think that it is not. I need to make an argument stronger that one of just cardinality, but I don't know what. I'm feel that I need to justify stuff in (*) but I'm not even sure if it is true! It took me a long time to think of this and I really need to get it done... help would be greatly appreciated.