9.3. Proof of Bezout’s Theorem

Try these first

1. 1.

   What does Bezout's Theorem state?

   Hint

   Think about relationships between two integers.

2. 2.

   Define the term GCD.

   Hint

   Consider what it means to evenly divide a number.

3. 3.

   What is the main assertion of Bezout's Theorem?

   • GCD can be expressed as multiplication
   • GCD can be expressed as a linear combination
   • GCD cannot be expressed

   Hint

   Think about how GCD relates to linear equations.

4. 4.

   True or False: The Extended Euclidean Algorithm finds the GCD and integer coefficients.

   • True
   • False

   Hint

   Consider the main functions of the Extended Euclidean Algorithm.

5. 5.

   Use integers a = 84 and b = 30. Find the GCD using both the Euclidean and Extended Euclidean methods, providing s and t.

   Hint

   Iterate through finding remainders until you reach zero.

6. 6.

   Suppose a and b are both even integers. Prove that their GCD is even using the principles from Bezout’s Theorem.

   Hint

   Consider the implications of linear combinations in your proof.