Question

Question 3 Find all solutions, if any, to [11] 246 + [45] 2011 = [4 in Z55. Express your answer in the form with 0 <1 < 55.

Answer 1

Solution : Given this problem we need to check itz. In order to solve the equation, satisties & [11] 246 +[us] " = u mod 55 - satisties the given cauation for all o 22455 2=0, we substitute given equation o [11] 046+ [us Jo" =4 mod 55 above equation = 0 44 mod 55 Hence 1-[0] does not satisby the #t=1, substitute equation o {17 146+ (us]," = 4 modss 11+ us =4 modus - 56=4174 mode 55 Hence x= [1] does not satisty the equation, we now do [11] x 46 + [us]x" mode 55 for various x x=2, substitute equation ④

{!!) qub £4532" 4 mod 55 113 (70368744177664) + Us (2008) = 4 mod 55 70368 704177664 + 92160 = 4 mod 55 = 474056 18 60 46464 tu mod55 (22,40) (23,34) Hence x= [2] does not statisty the equation. on putting x= 3,4 ... 54 in the above equation we get the following output. (21,21) (31,31) (3,14) (12,3u) (32,50) (4,26) (14, 36) (24, 46) (5,5) ( 33, $44) (6,6) (15,15) (25,25) (34,1) (7,29) (16, 16) (26, 26) (35,35) (8,19) (17,39) (27,49) (36,36) (931) (18,29) (28,34) (37,0) (10,10) (19,2,1) (29,51) (20,20) ( 30,30) (40,40) (1,41) (42,9) (44,lu) (52,19) (43,54) (u8,u) (53,9) (40,n) (49,16) (54,21), (us,45) (50,50) (46,46) (51,51) (38, 4939,6)

a where the first term in the bracket gives the input = 3,4,5.... su and the second term gives the output to find that the output is [u] only when x= [37], {u8] : The solutions are x= [37], [us].