To find the Highest Common Factor (HCF) of the numbers 30, 75, 90, and 135, we'll first break each number down into its prime factors:

1. **Prime factorization:**
   - \( 30 = 2 \times 3 \times 5 \)
   - \( 75 = 3 \times 5 \times 5 \) or \( 3 \times 5^2 \)
   - \( 90 = 2 \times 3^2 \times 5 \)
   - \( 135 = 3^3 \times 5 \)

2. **Common factors:**
   - The common prime factors are \( 3 \) and \( 5 \).

3. **Finding the HCF:**
   - The lowest powers of the common prime factors:
     - \( 3 \) appears in all the numbers with a minimum power of \( 1 \).
     - \( 5 \) also appears in all the numbers with a minimum power of \( 1 \).

   So, the HCF is \( 3 \times 5 = 15 \).

**The HCF of 30, 75, 90, and 135 is 15.**

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. What is the LCM of 30, 75, 90, and 135?
2. How can the Euclidean algorithm be used to find the HCF of two numbers?
3. What is the difference between HCF and LCM?
4. Can the HCF of a set of numbers be greater than any of the numbers?
5. How do you find the HCF of numbers that include negative values?
6. What is the HCF of a prime number and a composite number?
7. How do you find the HCF of decimal numbers?
8. What is the role of HCF in simplifying fractions?

**Tip:** When dealing with large numbers, breaking them down into prime factors can simplify finding the HCF or LCM.

Learn how to find the Highest Common Factor (HCF) of 30, 75, 90, and 135 with a detailed step-by-step solution. This problem covers key number theory concepts, including prime factorization and finding common factors. Suitable for students in Grades 6-8.

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