largest integer that divides both A and B.
Euclidean AlgorithmGCD
The Algorithm
The Euclidean Algorithm for finding GCD(A,B) is as follows:
- 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
通过递归的方法实现并检验：
class GCD{ public int GCD(int a, int b){ if(b == 0){ return a; } else{ return GCD(b, a%b); } } public static void main(String[] args) { GCD ob = new GCD(); int result = ob.GCD(270, 192); System.out.print(result); } }