There are three bells which ring after 30 min, 80 min and 90 min, respectively. If last time they rang simultaneously at 8:30 am. At what time they will again ringing simultaneously?

Options

A. `8:35` am
B. `8:35` pm
C. `8:30` pm
D. `8:40` am

Answer

To find out when the three bells will ring simultaneously again after ringing together at 8:30 AM, we need to calculate the least common multiple (LCM) of the intervals at which they ring: 30 minutes, 80 minutes, and 90 minutes.

### Step-by-Step Solution:

**Step 1: Find the LCM of 30, 80, and 90.**

1. **Prime Factorization:**
- 30 = 2 × 3 × 5
- 80 = 2^4 × 5
- 90 = 2 × 3^2 × 5

2. **Identify the highest power of each prime factor:**
- For 2: The highest power is 2^4 (from 80).
- For 3: The highest power is 3^2 (from 90).
- For 5: The highest power is 5^1 (from all three).

3. **Calculate the LCM:**
- LCM = 2^4 × 3^2 × 5
- LCM = 16 × 9 × 5
- LCM = 720

**Step 2: Convert LCM from minutes to hours.**

- Since 720 minutes is equal to hours, we divide by 60:
- 720 ÷ 60 = 12 hours

**Step 3: Add the LCM to the initial time.**

- The last time they rang together was at 8:30 AM.
- Adding 12 hours to 8:30 AM:
- 8:30 AM + 12 hours = 8:30 PM

### Final Answer:
The bells will ring together again at **8:30 PM**.

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