Count numbers up to C that can be reduced to 0 by adding or subtracting A or B
Given three non-negative integers A, B, and C, the task is to count the numbers in the range [1, C] that can be reduced to 0 by adding or subtracting A or B.
Input: A = 2, B = 4, C = 7
Explanation: The numbers from the range [1, 7] that can be reduced to 0 by given operations are:
- For element 2: The number can be modified as 2 – 2 = 0.
- For element 4: The number can be modified as 4 – 2 – 2 = 0.
- For element 6: The number can be modified as 6 – 4 – 2 = 0.
Therefore, the total count is 3.
Input: A = 2, B = 3, C = 5
Approach: The given problem can be solved based on the following observations:
- Consider X and Y number of addition or subtraction of A and B are performed respectively.
- After applying the operations on any number N, it becomes Ax + By. Therefore, by Extended Euclidean Algorithm, it can be said that there exist integer coefficients x and y such that Ax + By = GCD(A, B).
- Therefore, N must be a multiple of GCD(A, B), say G. Now the problem is reduced to finding the number of multiples of G which are in the range [1, C] which is floor (C / G).
Follow the below steps to solve the problem:
- Find the GCD of A and B and store it in a variable, say G.
- Now, the count of numbers over the range [1, C] is the multiples of G having values at most C which is given by floor(C/G).
Below is the implementation of the above approach:
Time Complexity: O(log(min(A, B)))
Auxiliary Space: O(1)