User:DAA 5N2/sandbox

euclid 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 GCD(270,192) = 6