Factors, Multiples and Prime Numbers

Factors

Factors of a number divide into it exactly. Factors of 12: 1, 2, 3, 4, 6, 12.

Multiples

Multiples are the "times table" of a number. Multiples of 5: 5, 10, 15, 20, 25, ...

Prime Numbers

A prime number has exactly two factors: 1 and itself. First primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

Note: 1 is NOT prime (only one factor). 2 is the only even prime.

Prime Factor Decomposition

Break a number into a product of primes using a factor tree:

$60 = 2^2 \times 3 \times 5$

HCF (Highest Common Factor)

The largest factor shared by two or more numbers.

Method: Write prime factorisations. Multiply the common prime factors (using the lowest power).

$\text{HCF}(24, 36)$: $24 = 2^3 \times 3$, $36 = 2^2 \times 3^2$. HCF = $2^2 \times 3 = 12$.

LCM (Lowest Common Multiple)

The smallest number that is a multiple of both.

Method: Multiply all prime factors (using the highest power).

$\text{LCM}(24, 36) = 2^3 \times 3^2 = 72$.

Venn Diagram Method

Place prime factors in a Venn diagram. The intersection contains shared factors.