You mean Z10/Z2, don't you? Or perhaps you are using a different notation than I am use to. Z10 can be characterized as 10 equivalence classes, labeled 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Modulo Z2 removes the "odd" labeled classes.
For the second part, note that 5 is prime so Z5 has no "zero-divisors". In Z10, 2(5)= 0 so 2 and 5 are "zero-divisors". What about "modulo 2"?
