To find the Highest Common Factor (HCF) of the numbers 560, 980, and 1456, we can use the method of prime factorization.

### Step 1: Prime Factorization of Each Number

1. **Prime factorization of 560**:
   - \( 560 \div 2 = 280 \)
   - \( 280 \div 2 = 140 \)
   - \( 140 \div 2 = 70 \)
   - \( 70 \div 2 = 35 \)
   - \( 35 \div 5 = 7 \)
   - \( 7 \div 7 = 1 \)
   - So, \( 560 = 2^4 \times 5^1 \times 7^1 \)

2. **Prime factorization of 980**:
   - \( 980 \div 2 = 490 \)
   - \( 490 \div 2 = 245 \)
   - \( 245 \div 5 = 49 \)
   - \( 49 \div 7 = 7 \)
   - \( 7 \div 7 = 1 \)
   - So, \( 980 = 2^2 \times 5^1 \times 7^2 \)

3. **Prime factorization of 1456**:
   - \( 1456 \div 2 = 728 \)
   - \( 728 \div 2 = 364 \)
   - \( 364 \div 2 = 182 \)
   - \( 182 \div 2 = 91 \)
   - \( 91 \div 7 = 13 \)
   - \( 13 \div 13 = 1 \)
   - So, \( 1456 = 2^4 \times 7^1 \times 13^1 \)

### Step 2: Identify the Common Prime Factors
The common prime factors among 560, 980, and 1456 are:
- \( 2 \)
- \( 7 \)

### Step 3: Choose the Lowest Power of Common Prime Factors
- The lowest power of \( 2 \) is \( 2^2 \).
- The lowest power of \( 7 \) is \( 7^1 \).

### Step 4: Multiply the Lowest Powers Together
- HCF \( = 2^2 \times 7^1 = 4 \times 7 = 28 \).

Thus, the HCF of 560, 980, and 1456 is **28**.

Would you like any details or further questions?

Here are five related questions to consider:
1. How do you find the Least Common Multiple (LCM) of the same three numbers?
2. Can the HCF of any two numbers ever be greater than the smaller of the two?
3. How does the Euclidean algorithm work for finding the HCF?
4. What is the relationship between the HCF and LCM of two numbers?
5. How can prime factorization be used to solve other types of problems?

**Tip:** Remember that the HCF of two or more numbers is the largest number that divides each of them without leaving a remainder.

Discover the step-by-step method to find the Highest Common Factor (HCF) of 560, 980, and 1456 through prime factorization. This detailed guide is ideal for students in Grades 6-8, covering key concepts in number theory and arithmetic.

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