Three bell sring at interval so 12, 18 and 36m in respectively. If they ring together at 11, 25 AM, when will they ring together for the first time after that?

Options

A. 11.56A.M
B. 12.04P.M
C. 12.01P.M
D. 12.50P.M

Answer

To solve the problem of when the three bells will ring together for the first time after 11:25 AM, we will follow these steps:

### Step 1: Find the LCM of the ringing intervals
The bells ring at intervals of 12, 18, and 36 minutes. We need to find the least common multiple (LCM) of these three numbers.

1. **Prime factorization**:
- 12 = 2² × 3¹
- 18 = 2¹ × 3²
- 36 = 2² × 3²

2. **Identify the highest powers of each prime**:
- For 2, the highest power is 2² (from 12 and 36).
- For 3, the highest power is 3² (from 18 and 36).

3. **Calculate the LCM**:
\[ \text{LCM} = 2^2 \times 3^2 = 4 \times 9 = 36 \]
So, the LCM of 12, 18, and 36 is 36 minutes.

### Step 2: Calculate the next time they will ring together
The bells will ring together again after 36 minutes from the time they last rang together at 11:25 AM.

1. **Add 36 minutes to 11:25 AM**:
- 11:25 AM + 36 minutes = 11:25 AM + 0:36 = 12:01 PM.

### Final Answer
The three bells will ring together for the first time after 11:25 AM at **12:01 PM**.

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