Question

Determine the Highest Common Factor (HCF) of each of the following:
A. 72 ; 108
72 = 2×2×3×3×3
90 = 2×3×3×5
324 = 2×2×3×3×3×3
Hcf =?
=?

HCF=?
=?

Answer 1

To determine the Highest Common Factor (HCF) of the numbers 72 and 108, we first find their prime factorizations.

### Prime Factorization:
- **72** can be factored as:
\[
72 = 2^3 \times 3^2
\]
(This is because \(72 = 2 \times 36 = 2 \times 2 \times 18 = 2 \times 2 \times 2 \times 9 = 2^3 \times 3^2\))

- **108** can be factored as:
\[
108 = 2^2 \times 3^3
\]
(This is because \(108 = 2 \times 54 = 2 \times 2 \times 27 = 2^2 \times 3^3\))

### Finding the HCF:
To find the HCF, we take the lowest power of all prime factors present in both numbers.
- For \(2\): The powers are \(3\) (from 72) and \(2\) (from 108). The lowest power is \(2\).
- For \(3\): The powers are \(2\) (from 72) and \(3\) (from 108). The lowest power is \(2\).

Thus, we calculate the HCF:
\[
\text{HCF} = 2^2 \times 3^2
\]

### Calculating the HCF:
Now, calculate \(2^2\) and \(3^2\):
\[
2^2 = 4
\]
\[
3^2 = 9
\]
So,
\[
\text{HCF} = 4 \times 9 = 36
\]

### Final Answer:
The Highest Common Factor (HCF) of 72 and 108 is **36**.

\[
\text{HCF} = 36
\]