GCD (Greatest Common Divisor) of two given numbers A and B is the highest number that can divide both A and B completely, i.e., leaving remainder 0 in each case. GCD is also called HCF(Highest Common Factor). There are various approaches to find the GCD of two given numbers.
The GCD of the given two numbers A and B can be calculated using different approaches.
- General method
- Euclidean algorithm (by repeated subtraction)
- Euclidean algorithm (by repeated division)
Input: 20, 30 Output: GCD(20, 30) = 10 Explanation: 10 is the highest integer which divides both 20 and 30 leaving 0 remainder Input: 36, 37 Output: GCD(36, 37) = 1 Explanation: 36 and 37 don't have any factors in common except 1. So, 1 is the gcd of 36 and 37
Note: gcd(A, B) = 1 if A, B are co-primes.
In the general approach of computing GCD, we actually implement the definition of GCD.
- First, find out all the factors of A and B individually.
- Then list out those factors which are common for both A and B.
- The highest of those common factors is the GCD of A and B.
A = 20, B = 30 Factors of A : (1, 2, 4, 5, 10, 20) Factors of B : (1, 2, 3, 5, 6, 10, 15, 30) Common factors of A and B : (1, 2, 5, 10) Highest of the Common factors (GCD) = 10
It is clear that the GCD of 20 and 30 can’t be greater than 20. So we have to check for the numbers within the range 1 and 20. Also, we need the greatest of the divisors. So, iterate from backward to reduce computation time.
GCD = 10
Euclidean algorithm (repeated subtraction):
This approach is based on the principle that the GCD of two numbers A and B will be the same even if we replace the larger number with the difference between A and B. In this approach, we perform GCD operation on A and B repeatedly by replacing A with B and B with the difference(A, B) as long as the difference is greater than 0.
A = 30, B = 20 gcd(30, 20) -> gcd(A, B) gcd(20, 30 - 20) = gcd(20,10) -> gcd(B,B-A) gcd(30 - 20, 20 - (30 - 20)) = gcd(10, 10) -> gcd(B - A, B - (B - A)) gcd(10, 10 - 10) = gcd(10, 0) here, the difference is 0 So stop the procedure. And 10 is the GCD of 30 and 20
GCD = 10
Euclidean algorithm (repeated division):
This approach is similar to the repeated subtraction approach. But, in this approach, we replace B with the modulus of A and B instead of the difference.
A = 30, B = 20 gcd(30, 20) -> gcd(A, B) gcd(20, 30 % 20) = gcd(20, 10) -> gcd(B, A % B) gcd(10, 20 % 10) = gcd(10, 10) -> gcd(A % B, B % (A % B)) gcd(10, 10 % 10) = gcd(10, 0) here, the modulus became 0 So, stop the procedure. And 10 is the GCD of 30 and 20
GCD = 10
Euclid’s repeated division approach is most commonly used among all the approaches.
Attention reader! Don’t stop learning now. Get hold of all the important Java Foundation and Collections concepts with the Fundamentals of Java and Java Collections Course at a student-friendly price and become industry ready.