4. Let a and b be integers. (a) Prove that [a]n = [b]n if and only if a = b (mod n). (b) Prove that either [a]n[b]n = Ø or [a]n = [b]n.

30. Prove statement of Theorem : for all integers .

a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is divisible by 11 if and only if 11 divides a0-a1+a2-+(1)nan, when z is written in the form as described in the previous problem. a. Prove that 10n1(mod9) for every positive integer n. b. Prove that a positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. (Hint: Any integer can be expressed in the form an10n+an110n1++a110+a0 where each ai is one of the digits 0,1,...,9.)

26. Let be an integer. Prove that . (Hint: Consider two cases.)