40 children are standing in a circle and one of them (say child-1) has a ring. The ring is passed clockwise. Child-1 passes on to child-2, child-2 passes on to child-4, child-4 passes on to child-7 and so on. After how many such changes (including child-1) will the ring be in the hands of child-1 again ?

Correct Option: 15

There are 40 children arranged in a circle numbered 1 to 40 clockwise. The ring is passed according to the sequence of recipients: Child-1 passes to Child-2, Child-2 passes to Child-4, Child-4 passes to Child-7, and so on. Let us trace the position of the child holding the ring after each step (change): Start: Child-1 (0 changes, holding ring). Step 1: 1 + 1 = 2 (Child-2). Step 2: 2 + 2 = 4 (Child-4). Step 3: 4 + 3 = 7 (Child-7). Step 4: 7 + 4 = 11. Step 5: 11 + 5 = 16. Step 6: 16 + 6 = 22. Step 7: 22 + 7 = 29. Step 8: 29 + 8 = 37. Step 9: 37 + 9 = 46. Since there are 40 children in a circle, we take modulo 40. 46 mod 40 = 6 (Child-6). Step 10: 6 + 10 = 16 (Child-16). Step 11: 16 + 11 = 27 (Child-27). Step 12: 27 + 12 = 39 (Child-39). Step 13: 39 + 13 = 52 => 52 mod 40 = 12 (Child-12). Step 14: 12 + 14 = 26 (Child-26). Step 15: 26 + 15 = 41 => 41 mod 40 = 1 (Child-1). Thus, after 15 changes, the ring returns to Child-1. Option (b) is correct.