The Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of primes, and this factorisation is unique (apart from the order of factors).

$$n = p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}$$

Example

Find the prime factorisation of $1260$.

$$1260 = 2^2 \times 3^2 \times 5 \times 7$$

Application: Finding LCM and HCF

For two numbers $a$ and $b$:

$$\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$$

Example: Find HCF and LCM of $12$ and $18$.

$$12 = 2^2 \times 3, \quad 18 = 2 \times 3^2$$ $$\text{HCF} = 2^1 \times 3^1 = 6$$ $$\text{LCM} = 2^2 \times 3^2 = 36$$

Verification: $6 \times 36 = 216 = 12 \times 18$ ✓