Show Isomorphism of ZxZ/<(a,b)> to Z if gcd(a,b)=1

[y2kevin]

Homework Statement

Show that ZxZ/<(a,b)> is isomorphic to Z if gcd(a,b)=1.

The Attempt at a Solution

I thought I had an idea but apparently I don't.

I reasoned this geometrically. For ZxZ/<(1,a)> (for all a in Z) can be graphed as a line hitting points (k,a*k) in the x-y plane. If we shift the line covered by <(1,a)> along the y-axis (e.g. use cosets (0,y)+<(1,a)>, where y is an integer), we can hit all points in ZxZ. Hence, ZxZ/<(1,a)> is isomorphic to Z.

But for ZxZ/<(2,3)>, the story changes, we skip all the (1,y) values if we shift by (0,y)+<(2,3)>. Hence, this leads me to conclude that we also need to include the possible cosets of (1,y)+<(2,a)>, making ZxZ/<(2,3)> isomorphic to Z₂xZ. But apparently this is wrong.

Can anyone shed a light on this? Thank you.

 

[y2kevin]

any ideas?