1. The problem statement, all variables and given/known data In Z, let H=<5> and K=<7>. Prove that Z=HK. Is Z the internal direct product of <5> and <7>? 2. Relevant equations 3. The attempt at a solution Since 3(5)-2(7)=1, every integer n element of Z can be written as 3n(5)+(-2n)(7), and n is an element of HK( where group operation is addition and powers written as multiples). Therefore Z=HK. Since the intersection of H and K =<35>, then Z is not equal to H direct product K. I think this is right but I am not sure on some of the justification of my claims. Can any one help me with the justification?