Six bells commence tolling together and toll at intervals of 2, 4, 6, 8 10 and…

Step-by-Step Calculation
1. Find the LCM of the intervals:
The intervals are 2, 4, 6, 8, 10, and 12 seconds.

Prime factorization:

2 = 2

4 = 2^2

6 = 2 * 3

8 = 2^3

10 = 2 * 5

12 = 2^2 * 3

Taking the highest power of each prime factor present (2^3, 3^1, 5^1):

LCM = 8 * 3 * 5 = 120 seconds.

This means the bells toll together every 120 seconds, or every 2 minutes.

2. Calculate the number of intervals in the total duration:

Total time = 30 minutes.

Number of intervals = Total time / Interval time = 30 / 2 = 15 intervals.

3. Account for the initial toll:

Since the bells start by tolling together at time zero, we must add that initial event to the 15 intervals that occur during the 30-minute duration.

Total tolls = 15 + 1 = 16 times.