The L.C.M. of 23 × 32 × 5 × 11, 24 × 34 × 52
 × 7 and 25 × 33 × 53
 × 72
 ×11 is:

To find the Least Common Multiple (L.C.M.) of the given numbers:

1. **First number**: \( 23 \times 32 \times 5 \times 11 \)
2. **Second number**: \( 24 \times 34 \times 52 \times 7 \)
3. **Third number**: \( 25 \times 33 \times 53 \times 72 \times 11 \)

The L.C.M. is found by taking the highest powers of all the prime factors involved.

### Prime Factor Breakdown:
- **Prime 2**: 
  - First number: \( 2^3 \)
  - Second number: \( 2^4 \)
  - Third number: \( 2^5 \)
  - Maximum power: \( 2^5 \)

- **Prime 3**: 
  - First number: \( 3^2 \)
  - Second number: \( 3^4 \)
  - Third number: \( 3^3 \)
  - Maximum power: \( 3^4 \)

- **Prime 5**: 
  - First number: \( 5 \)
  - Second number: \( 5^2 \)
  - Third number: \( 5^3 \)
  - Maximum power: \( 5^3 \)

- **Prime 7**: 
  - First number: No 7
  - Second number: \( 7^1 \)
  - Third number: \( 7^2 \)
  - Maximum power: \( 7^2 \)

- **Prime 11**: 
  - First number: \( 11^1 \)
  - Second number: No 11
  - Third number: \( 11^1 \)
  - Maximum power: \( 11^1 \)

### L.C.M. is the product of the highest powers of all primes:
\[
\text{L.C.M.} = 2^5 \times 3^4 \times 5^3 \times 7^2 \times 11^1
\]

Now, let's calculate the L.C.M:

\[
2^5 = 32, \quad 3^4 = 81, \quad 5^3 = 125, \quad 7^2 = 49, \quad 11^1 = 11
\]

The L.C.M. is:

\[
32 \times 81 \times 125 \times 49 \times 11
\]

Let me calculate the exact value.The Least Common Multiple (L.C.M.) of the given numbers is \( 174,636,000 \).

Would you like further details or explanations on this solution?

Here are 5 related questions to expand your understanding:
1. How do we find the greatest common divisor (GCD) of similar numbers?
2. What is the difference between LCM and GCD?
3. How does the LCM change if additional prime factors are introduced?
4. How would you find the LCM of more than three numbers?
5. Can the LCM be smaller than any of the given numbers?

**Tip:** The LCM helps in finding common periods or cycles in problems involving multiple repeating events.

The L.C.M. of 23 × 32 × 5 × 11, 24 × 34 × 52 × 7 and 25 × 33 × 53 × 72 ×11 is:

Learn how to find the Least Common Multiple (LCM) of large numbers using prime factorization. This problem breaks down the LCM of 23 × 32 × 5 × 11, 24 × 34 × 52 × 7, and 25 × 33 × 53 × 72 ×11 step by step. Explore key concepts in number theory and prime factorization, suitable for Grades 9-12.

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