Question

I have a problem in computing (i.e. classify) a factor group.

For example The group Z*Z*Z/<(3,6,9)> is isomorphic to Z_3*Z*Z. I can show this by contructing a homomorphism f

f(a,b,c) = ( [a]_3 a mod 3 , 2*a - b, 3*a - c )

and then show that Ker(f) = <(3,6,9)>. It is not hard to see that Im(f) = Z_3*Z*Z.

But how would I compute e.g. Z*Z/<(9,12)> ?

I guess I could create a function f(a,b) = ( a mod 9, 4*a - 3*b ).

then Ker(f) = <(9,12)>, but what is the image?