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To find the highest common factor (HCF) of the two numbers \( 2^3 \times 3^2 \times 5 \) and \( 2^2 \times 3^4 \times 5^3 \), follow these steps:

1. **Identify the common prime factors:** 
   - The prime factors common to both numbers are 2, 3, and 5.

2. **Take the lowest power of each common prime factor:**
   - For 2: The lowest power is \( 2^2 \) (since the powers are 3 and 2).
   - For 3: The lowest power is \( 3^2 \) (since the powers are 2 and 4).
   - For 5: The lowest power is \( 5^1 \) (since the powers are 1 and 3).

3. **Multiply these lowest powers together to get the HCF:**
   \[
   HCF = 2^2 \times 3^2 \times 5^1
   \]
   \[
   HCF = 4 \times 9 \times 5 = 180
   \]

So, the HCF of \( 2^3 \times 3^2 \times 5 \) and \( 2^2 \times 3^4 \times 5^3 \) is **180**.

Would you like more details, or do you have any questions about the solution?

Here are some related questions you might find helpful:
1. How do you find the Least Common Multiple (LCM) of two numbers?
2. Can the HCF be greater than either of the original numbers?
3. What is the significance of prime factorization in finding HCF?
4. How does the HCF relate to the GCD (Greatest Common Divisor)?
5. Can two numbers have an HCF of 1? If so, what does that mean?
6. How does the HCF of fractions differ from whole numbers?
7. How do you find the HCF of more than two numbers?
8. What are some real-world applications of finding the HCF?

**Tip:** When finding the HCF using prime factorization, always choose the smallest power of each common prime factor.

Learn how to find the highest common factor (HCF) of two numbers using prime factorization. This problem involves numbers expressed in terms of their prime factors and teaches how to determine the lowest powers of common prime factors to find the HCF. Suitable for students in Grades 7-9.

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