Hi Can help me to solve this questions: Hint : The course is( Discrete mathemati

ID: 3766276 • Letter: H

Question

Can help me to solve this questions:

Hint : The course is( Discrete mathematics for Computer Science) And the book is (

(Discrete Mathematics and Its Applications by Kenneth H. Rosen)

Question 1)

Suppose that a and b are integers, a 11 (mod 19), and b 3 (mod 19). Find the integer c with 0 c 18 such that

a) c a b (mod 19).

b) c 2a2 + 3b2 (mod 19).

Question 2)

Find an inverse of a modulo m for each of these pairs of relatively prime integers. (only solve a and b)

a) a = 4, m = 9

b) a =15, m =104

Question 3)

Find the integer a such that

a) a 15 (mod 27) and 26 a 0.

b) a 24 (mod 31) and 15 a 15.

Question 4)

Use the Euclidean algorithm to find gcd(111, 201).

a) c a b mod 19 a b 11 3 8 mod 19. So we can choose c = 8

c 2a 2 + 3b 2 mod 19 2a 2 + 3b 2 2 · 112 + 3 · 3 2 269 3 + 14 19 3 mod 19. So we can choose c = 3.

201 = 111*1 + 90
111 = 90*1 + 21
90 = 21*4 + 6
21 = 6*3 + 3
6 = 3*2 + 0

thus gcd(111, 201).= 3 = 29*111 + -16*201