The extended Euclidean algorithm computes the gcd of two integers ro and r₁ as a linear combination of the inputs. gcd(ro, r₁) = Here s and t are integers known as the Bezout coefficients. They are not unique. = s · ro + t · r₁ The algorithm works like the standard Euclidean algorithm, except that at each stage the current remainder î¿ is expressed as a linear combination of the inputs. rį = sįro + t¿r₁. = This produces a sequence of numbers To, 1,,Tn-1, n where n 0 and gcd(ro, r₁) = Tn-1. Suppose that ro T₁ = 382. What is GCD(420,382)? What is s? Give the sequence ro, T1,..., Tn-1, în in the blank below. Enter your answer as a comma separated list of numbers. What is t? = 420 and

Someone used the Euclidean Algorithm to compute gcd(44,140) and found the following. Use this to write the gcd as a linear combination of 44 and 140. Show work.  140 = 3(44) + 8 44 = 5(8) + 4 8 = 2(4) + 0

(a) Use the Euclidean algorithm to find gcd(131,326). (b) Use the above to find a solution to 131x+326y=gcd(131,326) (c) Does 131 have an inverse modulo 326? If so, find a value in {0,1,2,3,…,325} that is an inverse. If not, explain why not?

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