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(Euclidean Algorithm) As mentioned in class, the Euclidean algorithm is useful for finding the GCD of two numbers A and B by producing the sequence R, by R1 = mod (A, B), R2 = mod (B, Ry), R, = mod (R„ 2, Rµ 1) for n > 2. The last non-zero number in the sequence is the GCD of A and B. euclid_gcd Function: Input variables: • a scalar representing A • a scalar representing B (which you may assume is not equal to A) Output variables: • a scalar representing the GCD of A and B • a vector of all iterates produced (this vector should include R1 and R2 as its first entries as well as the final zero entry) A possible sample case is: » [GCD, R] = euclid_gcd(32, 40) GCD = 8 R = 32 8 0 » [GCD, R] = euclid_gcd(100, 65) GCD = 5 R = 35 30 5 0

(Euclidean Algorithm) As mentioned in class, the Euclidean algorithm is useful for finding the GCD of two numbers A and B by producing the sequence R, by R1 = mod (A, B), R2 = mod (B, Ry), R, = mod (R„ 2, Rµ 1) for n > 2. The last non-zero number in the sequence is the GCD of A and B. euclid_gcd Function: Input variables: • a scalar representing A • a scalar representing B (which you may assume is not equal to A) Output variables: • a scalar representing the GCD of A and B • a vector of all iterates produced (this vector should include R1 and R2 as its first entries as well as the final zero entry) A possible sample case is: » [GCD, R] = euclid_gcd(32, 40) GCD = 8 R = 32 8 0 » [GCD, R] = euclid_gcd(100, 65) GCD = 5 R = 35 30 5 0

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**Euclidean Algorithm** The Euclidean algorithm is useful for finding the GCD (greatest common divisor) of two numbers \( A \) and \( B \) by producing the sequence \( R_n \) as follows: - \( R_1 = \text{mod} (A, B) \) - \( R_2 = \text{mod} (B, R_1) \) - \( R_n = \text{mod} (R_{n-2}, R_{n-1}) \), for \( n \ge 2 \). The last non-zero number in the sequence is the GCD of \( A \) and \( B \). **euclid_gcd Function:** - **Input variables:** - A scalar representing \( A \). - A scalar representing \( B \) (which you may assume is not equal to \( A \)). - **Output variables:** - A scalar representing the GCD of \( A \) and \( B \). - A vector of all iterates produced (this vector should include \( R_1 \) and \( R_2 \) as its first entries as well as the final zero entry). **Sample Cases:** 1. **Case 1:** ``` >> [GCD, R] = euclid_gcd(32, 40) GCD = 8 R = 32 8 0 ``` 2. **Case 2:** ``` >> [GCD, R] = euclid_gcd(100, 65) GCD = 5 R = 35 30 5 0 ```
Transcribed Image Text:**Euclidean Algorithm** The Euclidean algorithm is useful for finding the GCD (greatest common divisor) of two numbers \( A \) and \( B \) by producing the sequence \( R_n \) as follows: - \( R_1 = \text{mod} (A, B) \) - \( R_2 = \text{mod} (B, R_1) \) - \( R_n = \text{mod} (R_{n-2}, R_{n-1}) \), for \( n \ge 2 \). The last non-zero number in the sequence is the GCD of \( A \) and \( B \). **euclid_gcd Function:** - **Input variables:** - A scalar representing \( A \). - A scalar representing \( B \) (which you may assume is not equal to \( A \)). - **Output variables:** - A scalar representing the GCD of \( A \) and \( B \). - A vector of all iterates produced (this vector should include \( R_1 \) and \( R_2 \) as its first entries as well as the final zero entry). **Sample Cases:** 1. **Case 1:** ``` >> [GCD, R] = euclid_gcd(32, 40) GCD = 8 R = 32 8 0 ``` 2. **Case 2:** ``` >> [GCD, R] = euclid_gcd(100, 65) GCD = 5 R = 35 30 5 0 ```
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