Minimum operations required to make two numbers equal

Given two integers A and B. the task is to find the minimum number of operations required to make A and B equal. In each operation, either of the below steps can be performed: 
 

• Increment either A or B with its initial value.
• Increment both A and B with their initial value

Examples: 
 

Input: A = 4, B = 10 
Output: 4 
Explanation: 
Initially A = 4, B = 10 
Operation 1: Increment A only: A = A + 4 = 8 
Operation 2: Increment A only: A = A + 4 = 12 
Operation 3: Increment A only: A = A + 4 = 16 
Operation 4: Increment A and B: A = A + 4 = 20 and B = B + 10 = 20 
They are equal now.
Input: A = 7, B = 23 
Output: 22 
Explanation: 
Initially A = 7, B = 23 
Operation 1 - 7: Increment A and B: A = 56 and B = 161 
Operation 8 - 22: Increment A: A = 161 and B = 161 
They are equal now. 
 

Approach: This problem can be solved using GCD. 
 

1. If A is greater than B, then swap A and B.
2. Now reduce B, such that gcd of A and B becomes 1.
3. Hence the minimum operations required to reach equal value is (B - 1).

For example: If A = 4, B = 10: 
 

• Step 1: Compare 4 and 10, as we always need B as the greater value. Here already B is greater than A. So, now no swap is required.
• Step 2: GCD(4, 10) = 2. So, we reduce B to B/2. Now A = 4 and B = 5. 
  GCD(4, 5) = 1, which was the target.
• Step 3: (Current value of B - 1) will be the required count. Here, Current B = 5. So (5 - 1 = 4), i.e. total 4 operations are required.

Below is the implementation of the above approach. 
 

14

Time Complexity: O(log(max(A, B))

Auxiliary Space: O(log(max(A, B))