Four bells ring at intervals of 4, 6,8 and 14 seconds. The start ringing simultaneously at 12.00 O' clock . At what time will they again ring simultaneously ?

Options

A. 12 hrs. 2 min . 48 sec
B. 12 hrs. 3 min
C. 12 hrs. 3 min. 20 sec.
D. 12 hrs. 3 min. 44 sec.

Answer

To find out when the four bells will ring simultaneously again after starting at 12:00 O' clock, we need to calculate the Least Common Multiple (LCM) of their ringing intervals: 4 seconds, 6 seconds, 8 seconds, and 14 seconds.

### Step-by-Step Solution:

**Step 1: Find the prime factorization of each interval.**
- 4 = 2^2
- 6 = 2^1 × 3^1
- 8 = 2^3
- 14 = 2^1 × 7^1

**Step 2: Identify the highest power of each prime number present in the factorizations.**
- For the prime number 2, the highest power is 2^3 (from 8).
- For the prime number 3, the highest power is 3^1 (from 6).
- For the prime number 7, the highest power is 7^1 (from 14).

**Step 3: Calculate the LCM using the highest powers identified.**
LCM = 2^3 × 3^1 × 7^1
= 8 × 3 × 7

**Step 4: Perform the multiplication.**
- First, calculate 8 × 3 = 24.
- Then, calculate 24 × 7 = 168.

Thus, the LCM of 4, 6, 8, and 14 is 168 seconds.

**Step 5: Convert seconds into minutes and seconds.**
- 168 seconds = 2 minutes and 48 seconds (since 168 ÷ 60 = 2 remainder 48).

**Step 6: Add this time to the starting time of 12:00.**
- 12:00 + 2 minutes 48 seconds = 12:02:48.

### Final Answer:
The four bells will ring simultaneously again at **12:02:48**.

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