The L.C.M. of 23 × 32 × 5 × 11, 24 × 34 × 52
 × 7 and 25 × 33 × 53
 × 72
 ×11 is:
2
A) 23 × 32 × 5 B) 25 × 34 × 53
C) 23 × 34 × 5 D) 25 × 34 × 53
 × 72
 ×11

To find the Least Common Multiple (LCM) of three numbers, we look at the prime factorization of each number and take the highest power of each prime factor present in any of the numbers.

Given the numbers are in factored form, let’s break them down first:

1. **Number 1**: \( 2^3 \times 3^2 \times 5 \times 11 \)
2. **Number 2**: \( 2^4 \times 3^4 \times 5^2 \times 7 \)
3. **Number 3**: \( 2^5 \times 3^3 \times 5^3 \times 7^2 \times 11 \)

### Step-by-Step Process for LCM:

We now find the highest powers of each prime factor across the three numbers:

- **For 2**: The highest power is \( 2^5 \).
- **For 3**: The highest power is \( 3^4 \).
- **For 5**: The highest power is \( 5^3 \).
- **For 7**: The highest power is \( 7^2 \).
- **For 11**: The highest power is \( 11 \).

Thus, the LCM is:
\[
LCM = 2^5 \times 3^4 \times 5^3 \times 7^2 \times 11
\]

This matches option **D**: \( 2^5 \times 3^4 \times 5^3 \times 7^2 \times 11 \).

### Answer: 
The correct option is **D**.

Would you like any more details or have further questions?

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Here are 5 related questions for practice:
1. How do you find the LCM of two numbers?
2. What is the difference between GCD and LCM?
3. How can you use the prime factorization method to find the LCM of large numbers?
4. What is the LCM of 12, 15, and 18?
5. How do you calculate the LCM of fractions?

**Tip**: Always break down numbers into their prime factorizations to make LCM and GCD calculations easier!

The L.C.M. of 2^3 × 3^2 × 5 × 11, 2^4 × 3^4 × 5^2 × 7, and 2^5 × 3^3 × 5^3 × 7^2 × 11 is: A) 2^3 × 3^2 × 5 B) 2^5 × 3^4 × 5^3 C) 2^3 × 3^4 × 5 D) 2^5 × 3^4 × 5^3 × 7^2 × 11

Discover how to find the Least Common Multiple (LCM) of numbers expressed in their prime factorized forms. This example demonstrates the application of the LCM formula and the LCM theorem, perfect for students in Grades 8-9.

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