9. Lecture – 57: Properties of GCD and Bezout’s Theorem

Try these first

1. 1.

   What is the GCD of 24 and 36?

   Hint

   Use the prime factorization method.

2. 2.

   State Bezout’s theorem in your own words.

   Hint

   Think about what linear combinations mean.

3. 3.

   What is Bezout's theorem?

   • A method to find primes
   • A theorem about GCD and linear combinations
   • An algorithm to find the multiplicative inverse

   Hint

   Think about what the theorem involves.

4. 4.

   True or False: The multiplicative inverse of a number a modulo N exists if and only if GCD(a, N) = 1.

   • True
   • False

   Hint

   Consider the relationship between inverses and GCD.

5. 5.

   Prove that for any integers a and b, there exist integers x and y such that ax + by = gcd(a, b).

   Hint

   Use examples such as (252, 198) during your proof.

6. 6.

   Given a = 13 and N = 24, find the multiplicative inverse modulo N.

   Hint

   Use the extended Euclidean algorithm to compute this.