Definition

1. We can write a whole number greater than 1 as a product of two whole numbers. 
E.g. 18 = 1 x 18 = 2 x 9 = 3 x 6.
Hence, 1, 2, 3, 6, 9 and 18 are called factors of 18.

2. Factors of a number are whole numbers which multiply to give that number. 

3. The common factors of two numbers are the factors that the numbers have in common. 
E.g. Factors of 12: 1, 2, 3, 4, 6, 12
       Factors of 21: 1, 3, 7, 21
Hence the common factors of 12 and 21 are 1 and 3. 

4. When we multiply a number by a non-zero whole number, we get a multiple of the number. 
E.g. Multiplies of 3: 3, 6, 9, 12, 15, ... 
       Multiplies of 5: 5, 10, 15, 20, 25, ...

5. The common multiplie of two numbers is a number that is a multiple of both numbers. 
In the above example of multiples of 3 and 5, the first common multiple would be 15. 

Definitions 
1. A prime number is a whole number greater than 1 that has exactly two different factors, 1 and itself. 
E.g. 5 = 1 x 5. Since 5 has no other whole number factors other than 1 and itself, it is a prime number. 

2. A composite number is a whole number greater than 1 that has more than 2 different factors.
E.g. 6 = 1 x 6 and 2 x 3. Since there are four factors (1, 2, 3 and 6), 6 is a composite number.
In other words, all whole numbers greater than 1 that are not prime numbers are composite numbers. 

Note: 0 and 1 are neither prime nor composite numbers! 

3. Prime factors are factors of a number that are also prime. 
E.g. The factors of 18 are 1, 2, 3, 6, 9 and 18. The prime factors of 18 are 2 and 3. 

4. The process of expressing a composite number as the product of prime factors is called prime factorisation.

5. We can use either the factor tree or repeated division to express a composite number as a product of its prime factors. 

The table below shows all the prime and composite numbers from 1 to 100, with all prime numbers shaded. 

Methods
The following picture shows examples of using the Factor Tree and Repeated Division respectively to express 180 as a product of prime factors. 

Finding HCF/LCM via Prime Factorisation 
Highest Common Factor (HCF)
1. The largest common factor among the common factors of two or more numbers is called the highest common factor (HCF) of the given numbers. 
For example,
Factors of 12 are 1, 2, 3, 4, 6, and 12.
Factors of 18 are 1, 2, 3, 6, 9 and 18.
The common factors of 12 and 18 are 1, 2, 3 and 6.
Thus the HCF of 12 and 18 is 6. 

2. Another method to find the HCF of two or more numbers is by using prime factorisation, which is the more efficient way. 

3. We can also repeatedly divide the numbers by prime numbers to find the HCF. 

Lowest Common Multiple (LCM) 
1. The smallest common multiple among the common multiples of two or more numbers is called the lowest common multiple (LCM) of the given numbers. 
For example,
Multiples of 8: 8, 16, 24, 32, 40, 48, ...
Multiples of 12: 12, 24, 36, 48, 60, ...
The common multiples of 8 and 12 are 24, 48, ...
Thus the LCM of 8 and 12 is 24. 

2. Another method to find the LCM of 2 or more numbers is by using prime factorisation.

3. We can also find the LCM of 2 or more numbers by repeatedly dividing the numbers by their prime factors. 

The following video shows an example of how HCF can be found using prime factorisation: 

Worked Example (Finding LCM using Prime Factorisation)
The following video shows an example of how LCM can be found using prime factorisation: 

Worked Example (Finding HCF/LCM using the Ladder Method)
The following video shows an example of how HCF/LCM can be found using the Ladder Method: 

Worked Example 
View the following for yet another worked example: 

Definition and Rules 1
1. Squares: To square a number, multiply the number by itself.
2. Square roots: When multiplied by itself, gives the number. E.g. √7 x √7 = 7

The properties of square roots are as follows:

Definition and Rules 2
1. When a number is multiplied by itself thrice, the product is called the cube of the number.
E.g. 5 x 5 x 5 = 125 or 53 = 125, where 53 is read as "5 cubed".

2. 125 is the cube of 5 and 5 is the cube root of 125. 
E.g. 3√125 = 5.

3. The numbers whose cube roots are whole numbers are called perfect cubes.
E.g. 1, 8, 27, 64, ... are perfect cubes.