Practice Extended Euclid’s Algorithm (9.4) - Lecture – 57: Properties of GCD and Bezout’s Theorem

Practice: Extended Euclid’s Algorithm

Lesson 9.4Lecture – 57: Properties of GCD and Bezout’s Theorem

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6 sample questions

Question 1.
What is the GCD of 12 and 15?
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Find the largest integer that divides both numbers.
Question 2.
Can you express the GCD of 8 and 12 as a linear combination?
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Use integer coefficients to represent the combination.
3Question 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.
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Think about how we discussed expressing the GCD.
4Question 4.
True or False: The coefficients in Bezout's theorem are always positive.
- True
- False
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Consider the examples we went through.
Question 5.
Prove that if a is a multiple of N, then it cannot have a multiplicative inverse modulo N.
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Consider the definition of GCD.
Question 6.
Use the Extended Euclidean Algorithm to find the GCD of 101 and 10, and express it using Bezout's coefficients.
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Document each step in the algorithm carefully.

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