Question 2b: If 208 = 33 mod 25, then prove the following properties of Modular Congruence: (i) Addition Property of Modular Congruence: If a = b mod n, and a+b=c, then

[ (a mod n) + (b mod n) ] mod n = c mod n (3 marks) (ii) Multiplication Property of Modular Congruence: If a = b mod n, and axb=c, then [ (a mod n) x (b mod n) ] mod n = c mod n (3 marks) Note: You need to demonstrate step-by-step calculations/operations in the answer. No marks will be awarded for only writing the final answer. Question 2c: Find all Primitive Roots of 11 and show your detailed working. (6 marks) Note: You need to demonstrate step-by-step calculations/operations in the answer. No marks will be awarded for only writing the final answer. Page 1 of 3 January 2024 This assessment is subject to the University Assessment Regulations for Candidates CS2DDA Question 2d: Why is Euler's Totient function useful and how would you calculate it for the prime number 5 and composite number 15? (5 marks) Question 2e: Why is Bézout's Lemma or Bézout's Identity an existential statement and what is its Coprime Condition? (3 marks) Question 2f: How is Modular Multiplicative Inverse different from Multiplicative Inverse and why it is used in cryptography? (3 marks)