User talk:DAA2016M1

1.Euclid's Algorithm
The Euclidean Algorithm is a technique for quickly finding the GCD of two integers.

The Algorithm

''The Euclidean Algorithm for finding GCD(A,B) is as follows:

If A = 0 then GCD(A,B)=B, since the GCD(0,B)=B, and we can stop.

If B = 0 then GCD(A,B)=A, since the GCD(A,0)=A, and we can stop.

Write A in quotient remainder form (A = B⋅Q + R)

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

B ≠0

Use long division to find that 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

B ≠0

Use long division to find that 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

B ≠0

Use long division to find that 78/36 = 2 with a remainder of 6. We can write this as: 78 = 36 * 2 + 6

Find GCD(36,6), since GCD(78,36)=GCD(36,6)

A=36, B=6

A ≠0

B ≠0

Use long division to find that 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

B =0, GCD(6,0)=6

So we have shown:

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

Therefore, GCD(270,192) = 6