Question:

Finding the greatest common divisor (GCD) of two positive integers results in the time complexity of O(log N),
where N is the larger number of two inputs. Justify the statement.

Question

Question:

Finding the greatest common divisor (GCD) of two positive integers results in the time complexity of O(log N), 
where N is the larger number of two inputs. Justify the statement.

Accepted Answer

Answer:

The statement is correct. The time complexity of finding GCD is O(log N) 
because it uses the Euclidean Algorithm, which repeatedly reduces the problem size.


Explanation:


The Euclidean Algorithm works as follows:



gcd(a, b) = gcd(b, a % b)



In each step:


    The larger number is replaced by the remainder of division
    The values decrease rapidly



This reduction happens logarithmically. That means:



    The number of steps required is proportional to log N
    N is the larger of the two numbers



For example:



gcd(48, 18)
→ gcd(18, 12)
→ gcd(12, 6)
→ gcd(6, 0)



Here, the numbers reduce quickly in very few steps.


Conclusion:

    Each step reduces the problem size significantly
    The total number of steps is proportional to log N
    Therefore, time complexity = O(log N)

Question: Finding the greatest common divisor (GCD) of two positive integers results in the time complexity of O(log N), where N is the larger number of two inputs. Justify the statement.

Answer with Explanation

Question:

Finding the greatest common divisor (GCD) of two positive integers results in the time complexity of O(log N), where N is the larger number of two inputs. Justify the statement.