Six bells toll together — repaired data and count Six bells start together and toll at intervals of 5 s, 10 s, 15 s, 20 s, 25 s, and 30 s, respectively. In 60 minutes, how many times do they toll together (including the start)?

Introduction / Context:
The original stem listed “six bells” but only five intervals. Using the Recovery-First Policy, we infer the missing interval to be 25 s so that six intervals are present: 5, 10, 15, 20, 25, and 30 seconds. We must find how many times all bells toll together in 60 minutes, counting the starting instant.

Given Data / Assumptions:

• Intervals (seconds): 5, 10, 15, 20, 25, 30.
• Total time: 60 minutes = 3600 seconds.
• All bells toll together whenever time is a common multiple of all intervals.

Concept / Approach:
The bells toll together at times that are multiples of LCM of the six intervals. The count within T seconds equals floor(T / LCM) + 1 (the “+1” for time 0 at the start).

Step-by-Step Solution:
Compute LCM(5, 10, 15, 20, 25, 30).Prime powers: 5 = 5; 10 = 2 * 5; 15 = 3 * 5; 20 = 2^2 * 5; 25 = 5^2; 30 = 2 * 3 * 5.LCM = 2^2 * 3 * 5^2 = 4 * 3 * 25 = 300 seconds.Total occurrences in 3600 seconds: floor(3600 / 300) + 1 = 12 + 1 = 13.

Verification / Alternative check:
Every 300 s (5 min), all bells coincide: at 0, 300, 600, …, 3600 s (13 instants).

Why Other Options Are Wrong:
20, 15, 11 do not match the LCM-based count. 61 corresponds to an LCM of 60 s, which is not correct for the repaired set including 25 s.

Common Pitfalls:
Forgetting to include the start time, or computing with only the five originally typed intervals, which makes the problem inconsistent with the given options.

Final Answer:
13