Euclidean Algorithm Assignment Help

The Euclidean Algorithm is usually a technique for easily choosing the GCD (Greatest Common Divisor) associated with two integers.

The Algorithm

The Euclidean Algorithm with regard to obtaining GCD(A, B) is really as comes after:

1. If A = 0 then GCD(A, B) = B,
   • Since the GCD(0, B) = B, and we can stop.
2. If B = 0 then GCD(A, B) = A,
   • Since the GCD(A, 0) = A, and we can stop.
3. Write A in quotient remainder form (A = B⋅Q + R)
4. Find GCD(B, R) using the Euclidean Algorithm
   • Since GCD(A, B) = GCD(B, R)

Example: Find the GCD of 270 and 192

• A = 270, B = 192
• A ≠ 0 (A not equal to 0)
• B ≠ 0 (B not equal to 0)
• Make use of long division to get in which 270/192 = 1 with a remainder of 78. We can write this as: 270 = 192 * 1 + 78
• Find GCD(192, 78), since GCD(270, 192) = GCD(192, 78)

A = 192, B = 78

• A ≠ 0 (A not equal to 0)
• B ≠ 0 (B not equal to 0)
• Make use of long division to get in which 192/78 = 2 with a remainder of 36. We can write this as:
• 192 = 78 * 2 + 36
• Find GCD(78, 36), since GCD(192, 78) = GCD(78, 36)

A = 78, B = 36

• A ≠ 0 (A not equal to 0)
• B ≠ 0 (B not equal to 0)
• Make use of long division to get in which 78/36 = 2 with a remainder of 6. We could compose this kind of since:
• 78 = 36 * 2 + 6
• Find GCD(36, 6), since GCD(78, 36) = GCD(36, 6)

A = 36, B = 6

• A ≠ 0 (A not equal to 0)
• B ≠ 0 (B not equal to 0)
• Make use of long division to get in which 36/6 = 6 with a remainder of 0. We can write this as:
• 36 = 6 * 6 + 0
• Find GCD(6, 0), since GCD(36, 6) = GCD(6, 0)

A = 6, B = 0

• A ≠ 0 (A not equal to 0)
• B =0 (B equal to 0)
• GCD(6, 0) = 6

And so we have now found:

GCD(270, 192) = GCD(192, 78) = GCD(78, 36) = GCD(36, 6) = GCD(6, 0) = 6 GCD(270, 192) = 6