# 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.

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

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

Numbers   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}

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