Five bells ring together at the intervals of 3, 5, 8, 9 and 10 seconds. All the bells ring simultaneously at the same time. They will again ring simultaneously after:...


To determine when the five bells will ring simultaneously again, we need to find the least common multiple (LCM) of the intervals at which each bell rings: 3, 5, 8, 9, and 10 seconds.

### Steps to calculate the LCM:

1. **Prime factorization**:
   - \( 3 = 3 \)
   - \( 5 = 5 \)
   - \( 8 = 2^3 \)
   - \( 9 = 3^2 \)
   - \( 10 = 2 \times 5 \)

2. **LCM is obtained by taking the highest powers of all prime factors**:
   - The highest power of 2 is \( 2^3 \) (from 8).
   - The highest power of 3 is \( 3^2 \) (from 9).
   - The highest power of 5 is \( 5 \) (from both 5 and 10).

Thus, the LCM is:
\[
LCM = 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 72 \times 5 = 360 \, \text{seconds}.
\]

So, the bells will ring simultaneously again after **360 seconds** (or 6 minutes).

Would you like more details on the process or any questions? Here are five related questions to deepen your understanding:

1. How do you find the LCM of two numbers using prime factorization?
2. What is the relationship between the LCM and GCD of two numbers?
3. Can you find the LCM of three numbers without prime factorization?
4. How can the concept of LCM be applied in real-world problems?
5. What is the difference between LCM and HCF (Highest Common Factor)?

**Tip**: Always break down numbers into their prime factors for easier LCM calculation, especially when dealing with larger sets of numbers.

Five bells ring together at the intervals of 3, 5, 8, 9 and 10 seconds. All the bells ring simultaneously at the same time. They will again ring simultaneously after:

Discover how to find the least common multiple (LCM) of the intervals 3, 5, 8, 9, and 10 seconds. This problem illustrates the use of prime factorization to solve when the bells will ring simultaneously again. Suitable for students in Grades 6-8.

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