Show your work

1. True or false. Note that you must explain your work. It is not enough to just answer true or false.

(i) 14 = 4 mod 4

(ii) -12 = 21 mod 11

(iii) In the multiplication table of Zn, each element of Zn appears exactly once in each row and each column (except for the first row and first column)

(iv) For any [a]n and [b]n in Zn, [a]n ×n [b]n = [b]n ×n [a]n.

(v) Given an integer n ≥ 2, [a]n ×n [0]n = [a]n for any integer a.

2. Short answer questions. Note that you must show work. It is not enough to just give answer.

(i) Write out all the numbers in the congruence class [6]9.

(ii) Calculate [23]29 x 29[15]29 if possible, if not explain why

(iii) Calculate [9]11 (divide) 11[6]11 if possible, if not explain why

(iv) Calculate [9]10 (divide) 10[6]10 if possible. If not, explain why

(v) Find the congruence class [a]8 in Z8 such that [a]8 x 8[5]8= [1]8