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:
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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