Factors & Multiples

Factors

The factors of a number are all the numbers which divide into it exactly. Factors can be thought of as the “atoms” of the number, since factor pairs multiply together to make the number.

Every factor has a partner.

Example

\[ 24 = 1 \times 24 \] \[ 24 = 2 \times 12 \] \[ 24 = 3 \times 8 \] \[ 24 = 4 \times 6 \]

When algebra is involved, remember to consider all combinations.

\[ \begin{align*} x^2 - 9 &= (x-3)(x+3) \\ x^2 + 5x &= x(x+5) \end{align*} \]

Highest Common Factor (HCF)

The largest factor that two or more numbers have in common is called the HCF.

Example

Find the HCF of 12 and 32

factors 12 and 32

Alternatively,

\[ \begin{align*} 12 &= 2 \times 2 \times 3 \\ 32 &= 2 \times 2 \times 2 \times 2 \times 2 \end{align*} \] Common factors: \(2 \times 2\) \[ \text{HCF} = 4 \]

Example

Find the HCF of 45 and 27

factors 45 and 27

Alternatively,

\[ \begin{align*} 45 &= 3 \times 3 \times 5 \\ 27 &= 3 \times 3 \times 3 \end{align*} \] Common factors: \(3 \times 3\) \[ \text{HCF} = 9 \]

Example

Find the HCF of \(36x\) and \(12x^2\)

factors 36x and 12x^2

Alternatively,

\[ \begin{align*} 36x &= 2 \times 2 \times 3 \times 3 \times x \\ 12x^2 &= 2 \times 2 \times 3 \times x \times x \end{align*} \] Common factors: \(2 \times 2 \times 3 \times x\) \[ \text{HCF} = 12x \]

Prime Factors

Prime numbers can only be divided exactly by themselves or 1.

First fifteen prime numbers:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Split the number using a prime factor.
Continue until only primes remain.
Write the result as a product of primes.

Example

Write 88 as a product of its prime factors.

\[ 88 = 11 \times 2 \times 2 \times 2 = 11 \times 2^3 \]

Multiples

A multiple of a number can be exactly divided by that number. Multiples can be thought of as “building blocks” made by the number.

Examples

Multiples of 5: 5, 10, 15, 20, …
Multiples of 12: 12, 24, 36, 48, …
Multiples of \(3x^2\): \(3x^2, 6x^2, 9x^2, 12x^2, …\)

Multiples of Fractions

Multiples of \( \tfrac{1}{8} \): \( \tfrac{1}{8},\; \tfrac{2}{8},\; \tfrac{3}{8},\; \tfrac{4}{8},\; \tfrac{5}{8},\; \tfrac{6}{8},\; \tfrac{7}{8},\; \tfrac{8}{8},\; \tfrac{9}{8},\; \tfrac{10}{8} \)
Multiples of \( \tfrac{1}{5} \): \( \tfrac{1}{5},\; \tfrac{2}{5},\; \tfrac{3}{5},\; \tfrac{4}{5},\; \tfrac{5}{5},\; \tfrac{6}{5},\; \tfrac{7}{5},\; \tfrac{8}{5},\; \tfrac{9}{5},\; \tfrac{10}{5} \)
Multiples of \( \tfrac{1}{20} \): \( \tfrac{1}{20},\; \tfrac{2}{20},\; \tfrac{3}{20},\; \tfrac{4}{20},\; \tfrac{5}{20},\; \tfrac{6}{20},\; \tfrac{7}{20},\; \tfrac{8}{20},\; \tfrac{9}{20},\; \tfrac{10}{20} \)

Lowest Common Multiple (LCM)

The LCM of two or more numbers is the lowest number which is a multiple of all of them.

Example

Find the LCM of 6 and 8

Multiples of 6: 6, 12, 18, 24, 30, 36, …
Multiples of 8: 8, 16, 24, 32, 40, …

The LCM is 24.

Mixing

HCF of Fractions

The HCF (Highest Common Factor) of fractions is found using the rule:

\( \text{HCF of fractions} = \dfrac{\text{HCF of numerators}}{\text{LCM of denominators}} \)

Steps

Find the HCF of the numerators.
Find the LCM of the denominators.
Form a new fraction using: \( \dfrac{\text{HCF of numerators}}{\text{LCM of denominators}} \)