Find the elements of the direct product

[thetodd]

1. The problem statement, all variables and given/known data

Compute the direct sum Z_12 (+) U(10)

Z_24 is the group Z under addition modulo 12
U(10) is the group Z under multiplication modulo 10

3. The attempt at a solution
I have computed direct sums of Z_n groups before:

For example: Z_2 (+) Z_3 = {(0,0),(0,1),(0,2),(1,0),(1,1),(1,2)}

From this I would think I would follow a similar process but my textbook has the example:
U(8) (+) U(10) = {(1,1),(1,3),(1,7),(1,9),(3,1),(3,3),(3,7),(3,9),( 5,1),(5,3),(5,7),(5,9),(7,1),(7,3),(7,7),(7,9)}

What's going on here?

[morphism]

Maybe what you're missing is that U(10) is actually the group of invertibles in Z_10 under multiplication. Can you write down what the elements of U(10) are?

[thetodd]

ok.. so U(10) = {1,3,7,9} under multiplication mod 10
and Z_12 = {0,1,2,3,4,5,6,7,8,9,10,11} under addition mod 12