9.4. Extended Euclid’s Algorithm

Try these first

1. 1.

   What is the GCD of 12 and 15?

   Hint

   Find the largest integer that divides both numbers.

2. 2.

   Can you express the GCD of 8 and 12 as a linear combination?

   Hint

   Use integer coefficients to represent the combination.

3. 3.

   What does Bezout's theorem state?

   • The GCD of two integers can be expressed as a product.
   • The GCD of two integers can be expressed as a linear combination.
   • The GCD is always a prime number.

   Hint

   Think about how we discussed expressing the GCD.

4. 4.

   True or False: The coefficients in Bezout's theorem are always positive.

   • True
   • False

   Hint

   Consider the examples we went through.

5. 5.

   Prove that if a is a multiple of N, then it cannot have a multiplicative inverse modulo N.

   Hint

   Consider the definition of GCD.

6. 6.

   Use the Extended Euclidean Algorithm to find the GCD of 101 and 10, and express it using Bezout's coefficients.

   Hint

   Document each step in the algorithm carefully.