Three bells ring at intervals of 36 seconds, 40 seconds and 48 seconds respectively. They start ringing together at a particular time. They will ring together after every

Options

A. 6 minutes
B. 12 minutes
C. 18 minutes
D. 24 minutes

Answer

To find out when three bells will ring together again after starting at the same time, we need to calculate the Least Common Multiple (LCM) of their ringing intervals: 36 seconds, 40 seconds, and 48 seconds.

### Step-by-Step Solution:

**Step 1: Find the prime factorization of each number.**
- For 36:
- 36 = 2 × 18
- 18 = 2 × 9
- 9 = 3 × 3
- So, the prime factorization of 36 is \(2^2 \times 3^2\).

- For 40:
- 40 = 2 × 20
- 20 = 2 × 10
- 10 = 2 × 5
- So, the prime factorization of 40 is \(2^3 \times 5^1\).

- For 48:
- 48 = 2 × 24
- 24 = 2 × 12
- 12 = 2 × 6
- 6 = 2 × 3
- So, the prime factorization of 48 is \(2^4 \times 3^1\).

**Step 2: Identify the highest power of each prime factor.**
- For the prime number 2:
- The highest power is \(2^4\) (from 48).

- For the prime number 3:
- The highest power is \(3^2\) (from 36).

- For the prime number 5:
- The highest power is \(5^1\) (from 40).

**Step 3: Calculate the LCM using the highest powers.**
- LCM = \(2^4 \times 3^2 \times 5^1\)
- LCM = \(16 \times 9 \times 5\)

**Step 4: Perform the multiplication step-by-step.**
- First, calculate \(16 \times 9 = 144\).
- Then, calculate \(144 \times 5 = 720\).

So, the LCM of 36, 40, and 48 is 720 seconds.

**Step 5: Convert seconds to minutes.**
- To convert 720 seconds to minutes, divide by 60:
- \(720 \div 60 = 12\) minutes.

### Final Answer:
The three bells will ring together again after **12 minutes**.

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