Show the steps and intermediate results of applying the extended Euclidean algorithm to compute the GCD of 512 and 240.

Answer :

The GCD of 512 and 240 is 16

What is the Euclidean algorithm?

The Euclidean Algorithm for determining GCD(A,B) is as follows: If A = 0, then GCD(A,B)=B, because GCD(0,B)=B, and we can stop. If B = 0, then GCD(A,B)=A because GCD(A,0)=A, and we can stop. Write A in quotient remainder form (A = BQ + R). GCD(B,R) can be found using the Euclidean Algorithm because GCD(A,B) = GCD(B,R) (B,R)

steps:

Create a division problem in which an is greater than b.

a/b = c with R as the remainder Make the division. Then, replace a with b, then R, and repeat the division. Repeat the process until R = 0.

512 ÷ 240 = 2 R 32    (512 = 2 × 240 + 32)

240 ÷ 32 = 7 R 16    (240 = 7 × 32 + 16)

32 ÷ 16 = 2 R 0    (32 = 2 × 16 + 0)

When remainder R = 0, the GCF is the divisor, b, in the last equation. GCF = 16

Hence to conclude GCD is 16

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