Consider two cars A and B, running infinitely (either clockwise or anti-clockwise) on a circular road. Given the speed of both the cars a and b. If a or b is positive, indicate they are moving in clockwise, Else they are moving in the anti-clockwise direction. The task is to find the number of distinct points they will meet each other at.
Input : a = 1, b = -1 Output : 2 Explanation Car A is moving clockwise while Car B is moving anti-clockwise but their speeds are same, so they will meet at two points i.e at the starting point and diametrically corresponding opposite point on the road. Input : a = 1, b = 2 Output : 1
Let the circumference of the circular road be d.
Let the time taken by cars A and B be ta and tb respectively. Their relative speed is a – b.
A and B start from the starting point and after some time, they will meet at starting point again. This time can be calculated by the LCM of ta and tb. Within this time period, they may meet at certain points which needs to be found out. Observe that, after they meet at the starting point they keep on meeting at the same point.
Time taken to meet again at the starting point will be,
T1 = LCM(ta, tb) = LCM(d/a, d/b) = d/GCD(a, b)
Let them meet N times in the time period T1.
So, the time delay between their consecutive meets is, say T2 can be calculated as,
T2 = (T1 / N).
This time can be calculated by calculating the time taken to meet for the first time after they start.
So, the time taken by them to meet for the first time,
Therefore, T2 = (d / (a – b)).
Dividing T1 by T2, we get,
N = (T1 / T2) = ((a – b) / GCD(a, b))
Below is implementation for the above approach:
- Steps required to visit M points in order on a circular ring of N points
- Count distinct points visited on the number line
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- Number of Integral Points between Two Points
- Print a number strictly less than a given number such that all its digits are distinct.
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- Check whether a number is circular prime or not
- Maximum number of segments that can contain the given points
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- Number of triangles in a plane if no more than two points are collinear
- Number of triangles formed from a set of points on three lines
- Print all combinations of points that can compose a given number
- Number of horizontal or vertical line segments to connect 3 points
- Number of distinct subsets of a set
- Number of distinct integers obtained by lcm(X, N)/X
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