Six bells commence tolling together and toll at intervals of 2,4,6,8,10 and 12…
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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 — 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 =…
- A.
4
- B.
10
- C.
15
- D.
16
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Correct answer: D
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.