Q2. Let as E N and let d be the units digit of as. 33d 10 (a) Prove that E N U {0}. 33 (b) Prove that d + 7d E 0 (mod 23) if and only if a: E 0 (mod 23). (This result can be use to recursively test whether a natural number is divisible by 23.) Q4. Determine, with justication, the remainder when 62022 is divided by 77. Q5. Consider the function f : Z a Z585 dened by f(9:) = [616] [m]+[135]. For each of the following statements, indicate whether it is true or false. Then prove or disprove the statement. (a) For all my 6 Z, if f(3:) = y), then 33 = y. (b) For every [y] E Z585, there exists a: E Z, such that f(33) = [y]. Note: The statement in part (a) says that the function is onetoone and the statement in part (b) says that the function is onto. This information is not needed to prove or disprove these statements, but you may have seen these terms before. Q6. Let a E Z Suppose Jl/In = p1 X 102 X , - . >