Question

Determine the Highest Common Factor (HCF) of each of the following:
A. 504 ; 648
504 = 2×2×2×3×3×7
648 = 2×2×2×3×3×3×4
Hcf =?
=?

HCF=?
=?

Answer 1

To find the Highest Common Factor (HCF) of 504 and 648, we first need the prime factorization of each number.

### Given Factorizations:
1. **504**: \( 504 = 2^3 \times 3^2 \times 7^1 \)
2. **648**: \( 648 = 2^3 \times 3^4 \)

### Finding HCF:
The HCF is found by taking the lowest power of each common prime factor.

- **For the prime factor 2**:
- In 504, the power is \( 3 \) (i.e., \( 2^3 \))
- In 648, the power is also \( 3 \) (i.e., \( 2^3 \))
- Lower power = \( 2^3 \)

- **For the prime factor 3**:
- In 504, the power is \( 2 \) (i.e., \( 3^2 \))
- In 648, the power is \( 4 \) (i.e., \( 3^4 \))
- Lower power = \( 3^2 \)

- **For the prime factor 7**:
- In 504, the power is \( 1 \) (i.e., \( 7^1 \))
- In 648, the prime factor 7 is not present (i.e., \( 7^0 \))
- Lower power = \( 7^0 \)

### Combine the lowest powers:
Now multiply the common prime factors raised to their respective lowest powers:
\[
HCF = 2^3 \times 3^2 \times 7^0
\]

Calculating this:
\[
HCF = 2^3 \times 3^2 \times 1 = 8 \times 9 = 72
\]

Thus, the Highest Common Factor (HCF) of 504 and 648 is **72**.