What is the highest common factor (HCF) of the numbers 425 and 476?

Question

Accepted Answer

To determine the highest common factor (HCF) of the numbers 425 and 476, we need to find the largest number that divides both 425 and 476 without leaving a remainder. The correct answer is given as Option 3, which is 17. Let's justify why this is correct and why the other options are incorrect.

### Explanation for Option 3 (Correct Answer: 17)

1. **Prime Factorization Method**:
   - First, we perform the prime factorization of both numbers.
   - **425**: The prime factors of 425 are 5 and 17. Specifically, 425 = 5 × 5 × 17.
   - **476**: The prime factors of 476 are 2 and 17. Specifically, 476 = 2 × 2 × 7 × 17.
   - The common prime factor between 425 and 476 is 17.
   - Therefore, the highest common factor (HCF) is 17.

2. **Verification**:
   - Dividing both numbers by 17:
     - 425 ÷ 17 = 25
     - 476 ÷ 17 = 28
   - Both divisions result in whole numbers, confirming that 17 is a common factor.

### Explanation for Why Other Options are Incorrect

1. **Option 1 (4)**:
   - 4 is not a factor of 425 because 425 ÷ 4 does not result in a whole number.
   - Therefore, 4 cannot be a common factor of both numbers.

2. **Option 2 (5)**:
   - While 5 is a factor of 425 (since 425 = 5 × 85), it is not a factor of 476 because 476 ÷ 5 does not result in a whole number.
   - Hence, 5 is not a common factor of both numbers.

3. **Option 4 (51)**:
   - 51 is not a factor of either 425 or 476 because neither 425 ÷ 51 nor 476 ÷ 51 results in a whole number.
   - Therefore, 51 cannot be a common factor of both numbers.

In conclusion, the highest common factor of 425 and 476 is indeed 17, as justified by the prime factorization method and verification through division. Thus, Option 3 is the correct answer.