Question

Using the extended Euclidean algorithm, provide step-by-step calculations.

Answer 1

The extended Euclidean algorithm is used to find the GCD of two numbers. It involves several steps, including initializing variables, calculating quotients, and updating variables. The GCD is the value of a, and the coefficients x and y are x1 and y1, respectively.

Step-by-step explanation:

Extended Euclidean Algorithm

The extended Euclidean algorithm is used to find the greatest common divisor (GCD) of two numbers, as well as to find the coefficients that satisfy Bézout's identity.

Step-by-Step Calculations:

1. Initialize variables: a, b, x1, x2, y1, y2
2. Set a = larger number, b = smaller number
3. Set x1 = 1, x2 = 0, y1 = 0, y2 = 1
4. While b is not equal to 0:
   1. Calculate the quotient q = a/b
   2. Update variables:
      • a = b
      • b = remainder of a/b
      • temp = x1
      • x1 = x2
      • x2 = temp - q*x2
      • temp = y1
      • y1 = y2
      • y2 = temp - q*y2
5. The GCD is the value of a, and the coefficients x and y are x1 and y1, respectively.