Given a pair of positive numbers x and y. We repeatedly subtract the smaller of the two integers from greater one until one of the integers becomes 0. The task is to count number of steps to before we stop (one of the numbers become 0).
Input : x = 5, y = 13 Output : 6 Explanation : There are total 6 steps before we reach 0: (5,13) --> (5,8) --> (5,3) --> (2,3) --> (2,1) --> (1,1) --> (1,0). Input : x = 3, y = 5 Output : 4 Explanation : There are 4 steps: (5,3) --> (2,3) --> (2,1) --> (1,1) --> (1,0) Input : x = 100, y = 19 Output : 13
A simple solution is to actually follow the process and count the number of steps.
A better solution is to use below steps. Let y be the smaller of two numbers
1) if y divides x then return (x/y)
2) else return ( (x/y) + solve(y, x%y) )
If we start with (x, y) and y divides x then the answer will be (x/y) since we can subtract y form x exactly (x/y) times.
For the other case, we take an example to see how it works: (100, 19)
We can subtract 19 from 100 exactly [100/19] = 5 times to get (19, 5).
We can subtract 5 from 19 exactly [19/5] = 3 times to get (5, 4).
We can subtract 4 from 5 exactly [5/4] = 1 times to get (4, 1).
We can subtract 1 from 4 exactly [4/1] = 4 times to get (1, 0)
hence a total of 5 + 3 + 1 + 4 = 13 steps.
Below is implementation based on above idea.
Time Complexity: O(log(n))
Auxiliary Space: O(1)
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