Problem 2:

Solution:

First we find the greatest common divisor of 1819 and 3587 using euclidean algorithm

3587 = 1819 (1) + 1768

1819 = 1768 (1) + 51

1768 = 51 (34) + 34

51 = 34 (1) + 17

34 = 17 (2) + 0

=> greatest common divisor of 1819 and 3587 is 17

Now we find the values of x and y from 1819 x + 3587 y = g.

choose x = 0 and y = 1

1819 (0) + 3587 (1) = 3587 → equation 1.

choose x = 1 and y = 0

1819 (1) + 3587 (0) = 1819 → equation 2

multiply with the first quotient from the steps of euclidean algorithm to equation 2 and subtract it from equation 1. r_0 = 1

1819 (-1) + 3587 (1) = 1768 → equation 3

multiply r_1 = 1 to equation 3 and subract it from equation 2

1819 (2) + 3587 (-1) = 51 → equation 4

multiply r_2 = 34 to equation 4 and subract it from equation 3

1819 (-69) + 3587 (35) = 34 → equation 5

multiply r_3 = 1 to equation 5 and subract it from equation 4

1819 (71) + 3587 (-36) = 17

=> x = 71 and y = -36.

Problem 1:

Solution:

We use Euclidean Algorithm,

a. (7469, 2464) = ?

7469 = 2464 (3) + 77

2464 = 77(32) + 0

=> g.c.d of 7469 and 2464 is 32

b. (2689, 4001) = ?

4001 = 2689 (1) + 1312

2689 = 1312 (2) + 65

1312 = 64 (20) + 12

64 = 12 (5) + 5

12 = 5(2) + 2

5 = 2 (2) + 1

2 = 1 (2) + 0

=> g.c.d of 7469 and 2464 is 1

c. (2947, 3997) = ?

3997 = 2947(1) + 1050

2947 = 1050 (2) + 847

1050 = 847 (1) + 203

847 = 203 (2) + 35

203 = 35 (5) + 28

35 = 28 (1) + 7

28 = 7 (4) + 0

=> g.c.d of 2947 and 3997 is 7

d. (1109, 4999) = ?

4999 = 1109 (4) + 563

1109 = 563 (1) + 546

563 = 546 (1) + 17

546 = 17 (20) + 6

17 = 6 (2) + 5

6 = 5 (1) + 1

5 = 1(5) + 0

=> g.c.d of 1109 and 4999 is 1