three bells ring at intervals of 9 minutes 15 minutes and 21 minutes.the bells will next ring together at 11.00pm.find the time the bells had rang last together.

To find the time when the three bells last rang together, we need to determine the Least Common Multiple (LCM) of the intervals at which they ring: 9 minutes, 15 minutes, and 21 minutes. This will give us the interval at which all three bells ring together.

### Step 1: Find the LCM of 9, 15, and 21

1. **Prime factorization:**
   - \(9 = 3 \times 3 = 3^2\)
   - \(15 = 3 \times 5\)
   - \(21 = 3 \times 7\)

2. **Take the highest powers of all prime factors:**
   - For \(3\), the highest power is \(3^2\).
   - For \(5\), the highest power is \(5\).
   - For \(7\), the highest power is \(7\).

3. **LCM:**
   \[
   \text{LCM} = 3^2 \times 5 \times 7 = 9 \times 5 \times 7 = 315 \text{ minutes}
   \]

### Step 2: Convert 315 minutes to hours and minutes
\[
315 \text{ minutes} = 5 \text{ hours and } 15 \text{ minutes}
\]

### Step 3: Calculate the time the bells last rang together
If the bells will next ring together at 11:00 PM, we subtract the LCM interval (5 hours and 15 minutes):

\[
11:00 \text{ PM} - 5 \text{ hours and } 15 \text{ minutes} = 5:45 \text{ PM}
\]

Therefore, the bells last rang together at **5:45 PM**.

Would you like further details or explanations? Here are some related questions you might find interesting:

1. How do you find the LCM of three numbers using their prime factors?
2. What is the significance of finding LCM in real-world problems?
3. How would the answer change if one of the bell intervals were different?
4. What are some other methods to find the LCM besides prime factorization?
5. How does understanding LCM help in scheduling problems?

**Tip:** When dealing with time intervals and synchronization, always consider using the LCM to find the common time frames!

Three bells ring at intervals of 9 minutes, 15 minutes, and 21 minutes. The bells will next ring together at 11:00 PM. Find the time the bells had last rung together.

This problem involves determining the last time three bells that ring at intervals of 9, 15, and 21 minutes rang together. Using the LCM (Least Common Multiple) method and prime factorization, the solution provides a step-by-step calculation, making it suitable for students in Grades 6-8.

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