Proving Z=HK: Group Theory Problem Homework

[tyrannosaurus]

Homework Statement

In Z, let H=<5> and K=<7>. Prove that Z=HK. Is Z the internal direct product of <5> and <7>?

Homework Equations

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 anyone help me with the justification?

 

[Dick]

What part are you finding a hard time justifying? You've shown Z=HK and 5^2*7^2=35*35 so it's not a direct product. What else do you need?