Question

# 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

CAT 2021

A
##### 3
CORRECT ANSWER WRONG ANSWER
B
##### 4
CORRECT ANSWER WRONG ANSWER
C
##### 7
CORRECT ANSWER WRONG ANSWER
D
##### 5
CORRECT ANSWER WRONG ANSWER
Solution

Correct option is

(A)

Explanatory Answer

Let length of the track = x

time taken to meet for the first time

$x/relative&space;speed&space;=&space;x/6-b;&space;or&space;,&space;x/b-6$

Time taken for a lap for A= x/6

Time taken for a lap for B= x/6

so , total time to meet for first time at starting point = LCM of (x/6 , x/b) which is equal to = x/HCF(6,b)

hence , number of meeting points on track = time taken to meet at starting for first meet = Relative speed / HCF of (6,b)

so , 6-b/HCF(6,b)=2 : or , b-6/HCF(6,b) = 2

the three different values which satisy the eqations are b = 2, 10 18

so , answer is 3

Hence, A is the correct option