The highest common factor (HCF) is the largest number that divides exactly into two or more numbers.

The lowest common multiple (LCM) is the smallest number that is a multiple of two or more numbers.

Both HCF and LCM can be found using prime factor decomposition or other suitable methods such as listing factors or multiples.

Using prime factor decomposition, each number is written as a product of its prime factors in index form.

For example, consider the numbers 12 and 18.

Prime factor decomposition of 12:

$$
12 = 2^2 \times 3
$$

Prime factor decomposition of 18:

$$
18 = 2 \times 3^2
$$

To find the HCF, take the lowest power of each prime that appears in both numbers:

$$
\text{HCF} = 2^1 \times 3^1 = 6
$$

To find the LCM, take the highest power of each prime that appears in either number:

$$
\text{LCM} = 2^2 \times 3^2 = 36
$$

You should be able to:

• Find the HCF and LCM using prime factor decomposition
• Use other appropriate methods where suitable
• Apply HCF and LCM to problem-solving situations

These skills are essential when simplifying fractions and working with multiples and factors.