Two friends A and B simultaneously start running around a circular track . They run in the same direction. A travels at 6 m/s and B runs at b m/s. If they cross each other at exactly two points on the circular track and b is a natural number less than 30, how many values can b take?

Let track length be equal to T.
Time taken to meet for the first time = T/relativespeed = T/(6−b) or T/(b−6)
Time taken for a lap for A = T/6
Time taken for a lap for B = T/b
So, time taken to meet for the first time at the starting point = LCM (T/6,T/b) = T/HCF(6,b)
Number of meeting points on the track = Time taken to meet at starting point/Time taken for first meeting = Relative speed / HCF (6,b).
So, in essence we have to find values for b such that (6−b)/HCF(6,b) = 2 or (b−6)/HCF(6,b) = 2

The question is ” If two people cross each other at exactly two points on the circular track and b is a natural number less than 30, how many values can b take?”
b = 2, 10, 18 satisfy this equation. So, there are three different values that b can take.
Hence, the answer is 3.

Choice A is the correct answer.