That's not difficult at all! In any ring ab= a(b+ 0)= ab+ a0 so a0= 0 for all a. But 1a= a for all a as well.

Just do the calculations: 1*1= 1, 3*7= 21= 1, 7*3= 21= 1, 9*9= 81= 1 so those are units. 2, 4, 6, and 8 cannot be units because any multiple must be even. 5 cannot be a unit because any multiple must end in 0 or 5.Also in Z_10 is it right that the units are 1, 3, 7 and 9 with inverses 1, 7, 3, 9, respectively?

Thanks for your help!