# Factorisation and prime numbers

A number is a multiple of another number if it is the product of this number with an **integer**.

An integer is a divisor of a number if the ratio of the number with the integer is an **integer**. It is also called a factor.

$$\Tblue{6}$$ is a multiple of $$\Tred{2}$$, $$\Tred{2}$$ is a divisor of $$\Tblue{6}$$, since $$\displaystyle\frac{\Tblue{6}}{\Tred{2}} = 3$$ is an integer.

To rephrase, take two integers $$\Tblue{m}$$ and $$\Tred{d}$$. $$\Tblue{m}$$ is a **multiple** of $$\Tred{d}$$ and $$\Tred{d}$$ is a **divisor** of $$\Tblue{m}$$ if the ratio $$\displaystyle \frac{\Tblue{m}}{\Tred{d}}$$ is an integer.

$$\Tred{4}$$ is a factor of $$\Tblue{12}$$, because $$\displaystyle\frac{12}{\Tred{4}} = 3$$ is an integer

$$\Tred{8}$$ is not a factor of $$\Tblue{12}$$ because $$\displaystyle\frac{12}{\Tred{8}} = 1.5$$ is not an integer.

An integer can have many **factors**. Every number has itself and $$1$$ as a factor. No integer can have infinitely many factors.

The factors of $$\Tblue{12}$$ are $$1$$, $$2$$, $$3$$, $$4$$, $$6$$ and $$12$$.

A number has an infinite number of **multiples**.

The multiples of $$\Tred{6}$$ are $$6$$, $$12$$, $$18$$, $$24$$, $$30$$, etc.

A common factor is a factor that two numbers share.

$$4$$ is a common factor of $$8$$ and $$12$$.

$$6$$ is not a common factor of $$18$$ and $$15$$.

Two integers have a unique highest common factor (HCF). It is also known as their greatest common divisor (GCD).

The factors of $$35$$ are $$1$$, $$\Tred{5}$$, $$7$$ and $$35$$. The factors of $$15$$ are $$1$$, $$3$$ , $$\Tred{5}$$ and $$15$$. So the highest common factor of $$15$$ and $$35$$ is $$\Tred{5}$$.

The common factors of $$100$$ and $$70$$ are $$1$$, $$2$$, $$5$$ and $$\Tred{10}$$. So $$\HCF(100,70) = \Tred{10}$$.

It is possible for the highest common factor of two numbers to be $$1$$ or one of the numbers.

$$$ \HCF(20, 9) = \Tred{1},\quad \HCF(6,9) = \Tred{3},\quad\HCF(6,18) = \Tred{6}. $$$

A common multiple of two integers is a multiple of both numbers.

Common multiples of $$\Tblue{3}$$ and $$\Tgreen{7}$$ are $$\Tred{21}$$, $$42$$, $$63$$, etc.

$$15$$ is not a common multiple of $$\Tblue{5}$$ and $$\Tgreen{9}$$, because it is not a multiple of $$9$$.

Two integers have a unique lowest common multiple (LCM).

The LCM of $$\Tblue{12}$$ and $$\Tgreen{15}$$ is $$\Tred{60}$$ because $$\Tred{60} = \Tblue{12} \times 5 = \Tgreen{15} \times 4$$

The LCM of $$\Tblue{8}$$ and $$\Tgreen{120}$$ is $$\Tred{120}$$ because $$\Tred{120} = \Tblue{8}\times 15$$

The highest common factor of two numbers can be one of the numbers or the product of the numbers.

$$$ \LCM(\Tblue{6},\Tgreen{18}) = \Tred{18},\quad \LCM(\Tblue{6},\Tgreen{9}) = \Tred{18},\quad \LCM(\Tblue{20}, \Tgreen{9}) = \Tred{180}. $$$ **The product of the HCF and LCM is the product of the numbers.**

We have $$\HCF(\Tblue{15}, \Tgreen{10}) = 5$$, $$\LCM(\Tblue{15}, \Tgreen{10}) = 30$$ and $$$150 = \Tblue{15}\times \Tgreen{10} = \HCF(\Tblue{15}, \Tgreen{10})\LCM(\Tblue{15},\Tgreen{10}).$$$

A prime number is a positive integer that has no factors apart from one and itself. $$1$$ is not considered to be a prime number.

$$2$$, $$3$$ and $$5$$ are prime numbers.

$$6$$ is not a prime number because it has factors $$2$$ and $$3$$.

There are infinitely many prime numbers. The first ones are

$$$2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,\dots$$$ Every positive integer is the **product of prime numbers**. The list of prime factors of a number is called its prime factorisation. The prime factorisation of every number is unique.

The number of times that a prime factor appears is its multiplicity.

$$\Tblue{2}$$ has multiplicity $$\Tred{3}$$ in the factorisation of $$24 = \Tblue{2}^\Tred{3} \times \Tgreen{3}$$.

If the prime factorisation of a number is known, it is easy to find **all the factors** of that number. The factors are **all possible products** of the prime factors.

We have the prime factorisation $$$24 = \Tblue{2}^\Tred{3} \times \Tgreen{3}.$$$ So all the factors of $$24$$ are : $$1$$, $$\Tblue{2}$$, $$\Tblue{2}^\Tred{2} = 4$$, $$\Tblue{2}^\Tred{3} = 8$$, $$\Tgreen{3}$$, $$\Tblue{2}\times\Tgreen{3} = 6$$, $$\Tblue{2}^\Tred{2} \times\Tgreen{3} = 12$$ and $$\Tblue{2}^\Tred{3} \times\Tgreen{3} = 24$$.

We can use **prime factorisation** to find the **highest common factor (HCF)** and the **lowest common multiple (LCM)** of two numbers.

- Compute the
**prime factorisation**of each number. $$$ 90 = \Tblue{2} \times \Tred{3}\times \Tblue{3}\times \Tred{5}, \qquad 105 = \Tred{3} \times \Tred{5} \times \Tgreen{7}$$$ - The
**HCF**is the product of all the prime factors that the two numbers have in common. $$$ \LCM(90,105) = \Tred{3}\times \Tred{5} = 15$$$ - The
**LCM**is the product of all the common prime numbers**once**and all the remaining prime numbers $$$ \HCF(90,105) = \Tblue{2}\times \Tred{3}\times \Tblue{3}\times \Tred{5}\times \Tgreen{7} = 630$$$ The LCM can be computed as well by the formula $$$\LCM(m,n) = \frac{mn}{\HCF(m,n)}. $$$