Question

1. Using the theorem divisibility, prove the following
a) If a|b , then a|bc ∀a, b, c ∈ ℤ ( 5 marks)
b) If a|b and b|c , then a|c (5 marks)
2. Using any programming language of choice (preferably python), implement the
following algorithms
a) Modular exponentiation algorithm (10 marks)
b) The sieve of Eratosthenes (10 marks)
3. Write a program that implements the Euclidean Algorithm (10 marks)
4. Modify the algorithm above such that it not only returns the gcd of a and b but also
the Bezouts coefficients x and y, such that + = 1 (10 marks)
5. Let m be the gcd of 117 and 299. Find m using the Euclidean algorithm
6. Find the integers p and q , solution to 1002 + 71 = 7. Determine whether the equation 486 + 222 = 6 has a solution such that , ∈ If yes, find x and y. If not, explain your answer.
8. Determine integers x and y such that (421, 11) = 421 + 11.

Answer 1

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