Number Theory

Choose the Correct Answer

Correct Answer : Option b

Explanation
Let's break down the problem:
30 students are seated in a circle, numbered 0 to 29 (let's say Student-A is at position 0).
The book is passed as follows:
1st pass: 1 step ahead (to position 1)
2nd pass: 2 steps ahead (to position 3)
3rd pass: 3 steps ahead (to position 6)
4th pass: 4 steps ahead (to position 10)
nth pass: n steps ahead
We need to find after how many passes will the book return to Student-A (position 0).
Mathematical formulation
After n passes, the total number of steps taken is:
S=1+2+3+…+n=n(n+1)/2
We want S to be a multiple of 30 (since there are 30 students, and the book returns to the starting position after a multiple of 30 steps):
n(n+1)/2≡0(mod30)
So,
n(n+1)≡0(mod60)
(because multiplying both sides by 2 gives n(n+1) must be divisible by 60).
Find the smallest positive integer n such that n(n+1) is divisible by 60
Let’s check the options:
(a) n = 14
14×15=210
14×15=210
210/60 = 3.5 (Not divisible)
(b) n = 15
15×16=240
15×16=240
240/60 = 4 (Divisible)
(c) n = 16
16×17=272
16×17=272
272/60 ≈ 4.53 (Not divisible)
(d) n = 18
18×19=342
18×19=342
342/60 = 5.7 (Not divisible)
Final Answer
(b) 15
After 15 passes, the book returns to Student-A.

Hence, option b is correct.