Six bells commence tolling together and toll at intervals of 2,4,6,8,10 and 12…
2025
Six bells commence tolling together and toll at intervals of 2,4,6,8,10 and 12 seconds respectively. In 30 minutes, how many times do they toll together?
Answer: D. 16 — Concept: When several periodic events start together, they recur simultaneously at time intervals equal to the LCM of their individual periods. Within a fixed…
- A.
4
- B.
10
- C.
15
- D.
16
Attempted by 5 students.
Show answer & explanation
Correct answer: D
Concept: When several periodic events start together, they recur simultaneously at time intervals equal to the LCM of their individual periods. Within a fixed duration, the number of simultaneous occurrences equals (duration ÷ LCM) + 1 — the initial moment itself counts as the first occurrence.
List the six tolling intervals: 2, 4, 6, 8, 10 and 12 seconds.
Find their LCM by prime factorisation: 2 = 21, 4 = 22, 6 = 21×31, 8 = 23, 10 = 21×51, 12 = 22×31. Taking the highest power of each prime gives LCM = 23×31×51 = 120 seconds.
Convert this cycle to minutes: 120 seconds = 2 minutes, so all six bells toll together every 2 minutes.
The given duration is 30 minutes, so the number of complete 2-minute cycles in it is 30 ÷ 2 = 15.
Add the initial simultaneous toll at time zero (the moment they all start together): 15 + 1 = 16.
Cross-check: Working entirely in seconds gives the same result — 30 minutes = 1800 seconds, and 1800 ÷ 120 = 15 cycles; adding the starting toll gives 15 + 1 = 16, confirming the count.