4. a. Use the Euclidean Algorithm to find the ged(816, 3612). 4. b. Use part 4a. to find the integers u and v such that according to Theorem 1.2 we have 816u+3612v = ged(816, 3612).

a) (123mod 19 ▪ 342 mod 19) mod 19 b) (12² mod 17)³ mod 13 2) (10 pts) Prove or disprove that for all integers a, b, and c, if alb and b/c then alc.

1. For each pair of integers m and n, use the division algorithm to divide m by n. Write the result in the form m = q⋅n + r where 0 < r < n. a. m = 32 and n = 9 b. m 62 and n = 7 = c. m -45 and n = 11 d. m 3 and n = 8 2. For each pair of integers a and b, use the Euclidean Algorithm to find the GCD(a, b). Then use Theorem 92 to find the LCM(a, b) a. a = 48 and b = 54 b. a = 330 and b = 156 c. a=1188 and b = 385 3. For each pair of integers m and d, calculate m mod d. Show your work! a. m 43 and d = 9 b. m 50 and d = 7 c. m 28 and d = 5 d. m = 30 and d = 2

To use the Egyptian​ algorithm, first rewrite the number 14as a sum of powers of two. 14=2+ ? +?