Study S1 Mathematics Maths - Prime factorisation - Geniebook

# Prime Numbers

## What is meant by Prime Numbers?

The numbers that have only two factors i.e. $$\textstyle 1$$ and $$\text{the number itself}$$ are known as Prime Numbers. So, there are $$25$$ prime numbers between $$1$$ and $$100$$, i.e.

$$\text{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}$$

Let’s understand this with the help of some examples:

#### Factors

$$1$$

$$1 × 1$$

$$1$$

$$2$$

$$1 × 2$$

$$1, 2$$

$$3$$

$$1 × 3$$

$$1, 3$$

$$4$$

$$1 × 4\\ 2 × 2$$

$$1, 2, 4$$

Look at the table above. Which numbers have exactly $$2$$ factors?

The answer would be $$2$$ and $$3$$.

So, a prime number is a whole number that has exactly $$2$$ factors, $$1$$ and itself.

Example: $$2, 3, 5, 7, 11, 13, 17$$ and so on.

## Composite Numbers

A composite number is a whole number that has more than two factors.

Example:

$$4, 6, 8, 9, 12, 14, 15$$ and so on.

Numbers

Factors

$$5$$

$$1 × 5$$

$$1, 5$$

$$6$$

$$1 \times 6\\ 2 \times 3$$

$$1, 2, 3, 6$$

$$7$$

$$1 × 7$$

$$1, 7$$

$$8$$

$$1 × 8\\ 2 × 4$$

$$1, 2, 4, 8$$

Look at the table above. Which numbers have more than $$2$$ factors?

The answer would be $$6$$ and $$8$$.

## Prime Factorisation

Example:

Express $$12$$ as a product of its prime factors.

Number   Prime Factor   Prime Factor   Prime Factor
$$12$$ $$=$$ $$2$$ $$\times$$ $$6$$
$$=$$ $$2$$ $$\times$$ $$2$$ $$\times$$ $$3$$

$$12 = 2 × 2 × 3$$ or $$12 = 2^2 × 3$$

Question 1:

Express $$175$$ as a product of its prime factors, leaving your answers in index notation.

Solution:

$$\begin{array}{c|lcr} 5 & 175 \qquad \\ \hline 5 & 35 \\ \hline 7 & 7 \\ \hline & 1 \end{array}$$

\begin{align*} 175 &= 5 × 5 × 7\\ &= 5^2 × 7 \end{align*}

## Square Roots And Cube Roots

To find the square root of a number, divide the index of each prime factor by 2.

Let’s understand this with the help of some examples:

Using a calculator, $$\sqrt{16} = 4$$; why?

Method 1:

 \begin{align*} 4 × 4 = 16\\ 4^2 = 16 \end{align*} \begin{align*} 16 &= 4^2\\ &= 4 \end{align*}

Method 2:

 \begin{align*} 2 × 2 × 2 × 2 = 16\\ 2^4 = 16 \end{align*} \begin{align*} \sqrt{16} &= 24\\ &= 2^2\\ &= 4 \end{align*}

To find the cube root of a number, divide the index of each prime factor by $$3$$

Let’s understand this with the help of some examples:

Question 2:

Using calculator, $$\sqrt[3]{64} = 4$$. Why?

Solution:

Method 1:

 \begin{align*} 4 × 4 × 4 = 64\\ 4^3 = 64 \end{align*} \begin{align*} \sqrt[3]{64} &= \sqrt[3]{4^3}\\ &= 4 \end{align*}

Method 2:

 \begin{align*} 2 × 2 × 2 × 2 × 2 × 2 = 64\\ 2^6 = 64 \end{align*} \begin{align*} \sqrt[3]{64} &= \sqrt[3]{2^6}\\ &=2^2\\ &= 4 \end{align*}

## Highest Common Factor (HCF)

Number Factors
$$12$$ $$1, 2, 3, 4, 6, 12$$
$$18$$ $$1, 2, 3, 6, 9, 18$$

The highest common factor of $$12$$ and $$18$$ is $$6$$.

To find the HCF of two or more numbers, multiply the common prime factors with the lowest index together.

Question 3:

Find the highest common factor (HCF) of $$55$$$$165$$ and $$605$$.

Solution:

Step 1: Prime factorization

\begin{align*} 55 &= 5 × 11\\ 165 &= 3 × 5 × 11\\ 605 &= 5 × 11^3 \end{align*}

Step 2: Identify common prime factors

$$5 \;\text{and} \;11$$

Step 3: Multiply the common prime factors with the lowest index.

\begin{align*} HCF &= 5 × 11\\ &= 55 \end{align*}

## Lowest Common Multiple (LCM)

Number Multiples
$$3$$ $$3, 6, 9, 12, 15, 18$$
$$4$$ $$4, 8, 12, 16, 20$$

The lowest common multiple of $$3$$ and $$4$$ is $$12$$.

To find the LCM of two or more numbers, multiply the unique prime factors with the highest index together.

Question 4:

Find the lowest common multiple (LCM) of $$18, 63 \;and \;81$$.

Solution:

Step 1: Prime factorization

\begin{align*} 18 &= 2 × 3^2\\ 63 &= 3^2 × 7\\ 81 &= 3^4 \end{align*}

Step 2: Identify the unique prime factors

$$2, 3, and \;7$$

Step 3: Find the highest index of each prime factor

\begin{align} \text{LCM} &= 2 × 3^4 × 7\\ &= 1134 \end{align}

Continue Learning
Basic Geometry Linear Equations
Number Patterns Percentage
Prime Numbers Ratio, Rate And Speed
Functions & Linear Graphs 1 Integers, Rational Numbers And Real Numbers
Basic Algebra And Algebraic Manipulation 1 Approximation And Estimation
Resources - Academic Topics
Primary
Primary 1
Primary 2
Primary 3
Primary 4
Primary 5
Primary 6
Secondary
Secondary 1
English
Maths
Basic Geometry
Linear Equations
Number Patterns
Percentage
Prime Numbers
Ratio, Rate And Speed
Functions & Linear Graphs 1
Integers, Rational Numbers And Real Numbers
Basic Algebra And Algebraic Manipulation 1
Approximation And Estimation
Science
Secondary 2
Secondary 3
Secondary 4
Sign up for a free demo
(P1 to S4 levels)
Our Education Consultants will get in touch to offer a complimentary product demo and Strength Analysis to your child.
Ready to power up your
child's academic success?
Let our Education Consultants show you how.
*By submitting your phone number, we have your permission to contact
you regarding Geniebook. See our Privacy Policy.