9.2. Bezout’s Theorem

Try these first

1. 1.

   What is Bezout's Theorem?

   Hint

   Think about how GCD relates to other integer properties.

2. 2.

   How do you express a linear combination?

   Hint

   Recall the defining equation involving GCD.

3. 3.

   What does Bezout's theorem state?

   • The GCD is always positive
   • The GCD can be expressed as a linear combination of its integers
   • The GCD is always an integer

   Hint

   Recall the relationship between GCD and linear combinations.

4. 4.

   True or False: If two integers are co-prime, their GCD is 1.

   • True
   • False

   Hint

   Think about the definition of co-prime.

5. 5.

   Use the extended Euclidean algorithm to find the GCD of 57 and 34. Express the GCD as a linear combination using Bezout's coefficients.

   Hint

   Track remainders and apply backward substitution.

6. 6.

   Determine if the integer 45 has a multiplicative inverse under modulo 64 and explain your reasoning.

   Hint

   Check the GCD result to confirm if an inverse exists.