Question

Determine the Highest Common Factor (HCF) of each of the following:
A. 112 ; 648
125 = 5×5×5
352 = 5×5×13
Hcf =?
=?

HCF=?
=?

Answer 1

To find the Highest Common Factor (HCF), we can factor the given numbers into their prime factors.

### A. Find the HCF of 112 and 648.

1. **Prime factorization of 112:**
- 112 is an even number, so we divide by 2:
- \( 112 \div 2 = 56 \)
- \( 56 \div 2 = 28 \)
- \( 28 \div 2 = 14 \)
- \( 14 \div 2 = 7 \)
- 7 is a prime number.
- So, the prime factorization of 112 is:
\[
112 = 2^4 \times 7^1
\]

2. **Prime factorization of 648:**
- 648 is also even, so we divide by 2:
- \( 648 \div 2 = 324 \)
- \( 324 \div 2 = 162 \)
- \( 162 \div 2 = 81 \)
- 81 is \( 3^4 \) because:
- \( 81 \div 3 = 27 \)
- \( 27 \div 3 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)
- So, the prime factorization of 648 is:
\[
648 = 2^3 \times 3^4
\]

3. **Find the HCF:**
- The common prime factors between 112 and 648 are:
- For \( 2 \): The minimum power is \( 2^3 \).
- There are no common factors of \( 3 \) or \( 7 \).
- Therefore, the HCF is:
\[
\text{HCF} = 2^3 = 8
\]

### B. Determine HCF of the factored numbers:

Given:
- \( 125 = 5^3 \)
- \( 352 = 5^2 \times 13^1 \)

4. **Find the HCF of 125 and 352:**
- The common prime factor is \( 5 \).
- For \( 5 \): The minimum power is \( 5^2 \).
- Therefore, the HCF is:
\[
\text{HCF} = 5^2 = 25
\]

### Summary:
- **HCF of 112 and 648 = 8**
- **HCF of 125 and 352 = 25**