Question

Please write neatly and clearly, show all work. Thank you! (I've been stumped for awhile)

(1 point) Find the smallest positive integer x that solves the congruence: 11x = 4 (mod 68) x = (Hint: From running the Euclidean algorithm forwards and backwards we get 1 = s(11) + +(68). Find s and use it to solve the congruence.)

Answer 1

Solution :- *) Given that The congruence is 112 = 4 (mod 68) -> (1) Here We Knoco that the linear congruence QX =b (mod n). has a Solution d16 Where d = ged (a,m) . From the given tinear congruence , we get Q=11, b=4, n=68 * d = ged (a,m) = ged (11,68) = 1 and * £ £ That is do Thus, the solution of linear congovence exist : NOCO We need to calculate the soletion of linear congruence Therefore For this substitute in the foom of 1 - 11Q + 686 Where.'a' will be inverse of 11.

NOCO bring the given congovence in the above from by esing Euclid algorithm let Euclid division algorithm : From the given integers @ and 'b' with b#0 , these exists a unique integer (q' and 'X' Satisfying a = bq + ,0<x< 161 → 68 = 6x11+2 -> (2) 11 = 5x9 +1 -> (3) NOCO From equ (8), we get + 1 = 11-(5 x2) = 11-(5x (68-6x 1)) [Since the value of igi is from equ (a) = 11 - 5x68 + 5X 6X U. = l +5X6 X 11 - 5X 68 = 11 (1+ 5x 6) - 5x68 = 11(31) – 5868)

1 = 11(31) + 68(-5) + 1 = 11a + 68b Thus the inverse of 11 is 31 :: Multiplying the equers by 31, we get → 34100 = 124 (mod 68) X = 194 (mod 68) Hence we can also written as X = 124** + 68 K Where ķ -0,1,9... → X = 124, 126 + 68X1, 124 + 68x9 ... 2. The smallest is 1241 * ** Thanking you ----