Prime Numbers, Composite Numbers and Prime Factorisation (Questions 1 – 4)
• A prime number is a whole number that has exactly two different factors, $1$ and itself.
• A composite number is a whole number that has more than two different factors.
• Special cases:
$\circ\ 1$ is neither a prime number nor a composite number;
$\circ\ 2$ is the first prime number, and the only prime number that is an even number.
• Prime factor(s) of a whole number is/are its factor(s) that is/are prime number(s).
• Method to determine whether a whole number is a prime number or not:
$\circ\ $ Test whether each prime number that is less than or equal to the square root of the given whole number is a factor of it or not. If none of these prime numbers is a factor of the given number, then this number is a prime number; Otherwise, it is a composite number.
• Prime factorisation of a whole number is the process of expressing the number as the product of its prime factors. If possible, the prime factorisation of a whole number is usually expressed in index notation.
Figure 1.1 is an example of finding the prime factorisation of $18$ using long division.
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Figure 1.1 Finding the Prime Factorisation of $18$ Using Long Division
Highest Common Factor and Lowest Common Multiple (Questions 5 – 8, 23 – 25 )
1. Methods to find the highest common factor (HCF) of two or more numbers:
$\circ\ $ By Prime Factorisation


Find the prime factorisation of each number in index notation;
Then the HCF of these numbers is the product of the common prime factors that are each to the lowest power in the prime factorisations.
$\circ\ $ By Long Division.
Start from dividing each number by the smallest common prime factors, until there are no common prime factors for all of the numbers.
Then the HCF of these numbers is the product of the common prime factors on the left.
2. Methods to find the lowest common multiple (LCM) of two or more numbers:
$\circ\ $ By Prime Factorisation:
Find the prime factorisation of each number in index notation;
Then the LCM of these numbers is the product of all of the prime factors that are each to the highest power in the prime factorisations.
$\circ\ $ By Long Division.
Start from dividing each number by the smallest common prime factors, until there are no common prime factors for any two of the numbers.

Then the LCM of these numbers is the product of the common prime factors on the left and all of the remaining factors.
Square Roots and Cube Roots Involving Prime Factorisation (Questions 9 – 14)
• A number that is the square of a whole number is known as a perfect square (or square number).
Correspondingly, a number whose square is a given number is known as the square root of the given number, denoted by the square root symbol “$\sqrt{{}}$”.
• A number that is the cube of a whole number is known as a perfect cube (or cube number).

Correspondingly, a number whose cube is a given number is known as the cube root of the given number, denoted by the cube root symbol “$\sqrt[3]{{}}$.
• Prime factorisation can be used to determine the square/cube root of a given number or determine whether a given number is a perfect square/cube or not.
Applications of HCF and LCM (Questions 15 – 22, 26)
Finding the HCF and LCM of numbers are applied under many real-world situations to evaluate the numerical property of data or provide us with quantitative reference in decision-making. For more specific real-world applications of HCF and LCM, see questions of this unit.

Prime Numbers, Composite Numbers and Prime Factorisation (Questions 1 – 4)

• A **prime number** is a whole number that has exactly two different factors, $1$ and itself.

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

$\circ\ 1$ is neither a prime number nor a composite number;

$\circ\ 2$ is the first prime number, and the only prime number that is an even number.

• **Prime factor(s)** of a whole number is/are its factor(s) that is/are prime number(s).

• Method to determine whether a whole number is a prime number or not:

$\circ\ $ Test whether each prime number that is less than or equal to the square root of the given whole number is a factor of it or not. If none of these prime numbers is a factor of the given number, then this number is a prime number; Otherwise, it is a composite number.

• **Prime factorisation** of a whole number is the process of expressing the number as the product of its prime factors. If possible, the prime factorisation of a whole number is usually expressed **in index notation.**

Figure 1.1 is an example of finding the prime factorisation of $18$ using long division.

Figure 1.1 Finding the Prime Factorisation of $18$ Using Long Division

Highest Common Factor and Lowest Common Multiple (Questions 5 – 8, 23 – 25 )

1. Methods to find the **highest common factor (HCF)** of two or more numbers:

Find the prime factorisation of each number in index notation;

Then the HCF of these numbers is the product of the common prime factors that are each to the lowest power in the prime factorisations.

Start from dividing each number by the smallest common prime factors, until there are no common prime factors for all of the numbers.

Then the HCF of these numbers is the product of the common prime factors on the left.

2. Methods to find the **lowest common multiple (LCM)** of two or more numbers:

Then the LCM of these numbers is the product of all of the prime factors that are each to the highest power in the prime factorisations.

Start from dividing each number by the smallest common prime factors, until there are no common prime factors for any two of the numbers.


Then the LCM of these numbers is the product of the common prime factors on the left and all of the remaining factors.

Square Roots and Cube Roots Involving Prime Factorisation (Questions 9 – 14)

• A number that is the square of a whole number is known as a **perfect square** (or **square number**).

Correspondingly, a number whose square is a given number is known as the square root of the given number, denoted by the square root symbol “$\sqrt{{}}$”.

• A number that is the cube of a whole number is known as a **perfect cube** (or **cube number**).


Correspondingly, a number whose cube is a given number is known as the **cube root** of the given number, denoted by the cube root symbol “$\sqrt[3]{{}}$.

• Prime factorisation can be used to determine the square/cube root of a given number or determine whether a given number is a perfect square/cube or not.

Applications of HCF and LCM (Questions 15 – 22, 26)

Finding the HCF and LCM of numbers are applied under many real-world situations to evaluate the numerical property of data or provide us with quantitative reference in decision-making. For more specific real-world applications of HCF and LCM, see questions of this unit.