Question

Show the significant steps of your work clearly for ALL problems. You may receive zero or reduced points for insufficient work. 1. Use the Euclidean Algorithm to find the greatest common divisor (gcd) of: 1591 and 629 and write this gcd as a linear combination of 1591 and 629.

Answer 1

Using the Euclidean Algorithm, the gcd of 1591 and 629 is found to be 37, which can be expressed as a linear combination: 37 = 1591 × 2 - 629 × 5.

Step-by-step explanation:

The goal is to use the Euclidean Algorithm to find the greatest common divisor (gcd) of 1591 and 629 and express this gcd as a linear combination of 1591 and 629.

1. Divide 1591 by 629 to find the quotient and remainder: 1591 = 629 × 2 + 333.
2. Next, divide 629 by the remainder (333): 629 = 333 × 1 + 296.
3. Continue this process with 333 and 296: 333 = 296 × 1 + 37.
4. Now divide 296 by 37: 296 = 37 × 8 + 0. At this point, since the remainder is 0, the gcd is the last non-zero remainder, which is 37.
5. To express 37 as a linear combination of 1591 and 629, we trace back our steps substituting remainders:

• We know from step 3 that 37 = 333 - 296 × 1.
• From step 2, we substitute 296: 37 = 333 - (629 - 333 × 1) × 1 which simplifies to 37 = 333 × 2 - 629.
• From step 1, we substitute 333: 37 = (1591 - 629 × 2) × 2 - 629 which simplifies to 37 = 1591 × 2 - 629 × 5.

This completes the process of finding the gcd of 1591 and 629 and expressing it as a linear combination of the two numbers.