Factors Of A Number

If a number can be expressed as a product of two whole numbers, then the whole numbers are called factors of that number.

So, the factors of 6 are 1, 2, 3 and 6.

Example 1

Find all factors of 45.

Solution:

So, the factors of 45 are 1, 3, 5, 9, 15 and 45.

Common Factors

10 = 2 × 5 = 1 × 10 Thus, the factors of 10 are 1, 2, 5 and 10.

15 = 1 × 15 = 3 × 5 Thus, the factors of 15 are 1, 3, 5 and 15.

Clearly, 5 is a factor of both 10 and 15. It is said that 5 is a common factor of 10 and 15.

Example 2

Find a common factor of: a. 6 and 8 b. 14 and 21

Solution:

Prime Numbers

If a number has only two different factors, 1 and itself, then the number is said to be a prime number.

For example, 31 = 1 × 31

31 is a prime number since it has only two different factors.

Note:

But 1 is not a prime number since it does not have two different factors.

Example 3

Express 150 as a product of prime numbers, i.e. find its prime factor decomposition.

Solution:

Note:

We try the prime numbers in order of their magnitude.

We observe that:

The prime factor decomposition of a number is unique.

This is called the Fundamental Theorem of Arithmetic. It provides us with a good reason for defining prime numbers so as to exclude 1. If 1 were a prime, then the prime factor decomposition would lose its uniqueness. This is because we could multiply by 1 as many times as we like in the decomposition.

Highest Common Factor

The highest common factor (HCF) of two numbers (or expressions) is the largest number (or expression) that is a factor of both.

Consider the highest common factor of 16 and 24.

The common factors are 2, 4 and 8. So, the highest common factor is 8.

Note:

The highest common factor is the product of the common prime factors.

In general:

To find the highest common factor of two (or more) numbers, make prime factors of the numbers and identify the common prime factors. Then the highest common factor is the product of the common prime factors.

Example 4

Find the highest common factor of 60 and 150.

Solution:

The prime factorisation of 60 is:

The prime factorisation of 150 is:

Note:

The highest common factor can also be obtained by a trial and error method.

For example, the highest common factor of 40 and 45 is 5 because 5 is the largest number which divides into both 40 and 45 exactly.

Likewise, the highest common factor of 27 and 36 is 9 because 9 is the largest number which divides into both 27 and 36 exactly.

Key Terms

factors, common factors, prime numbers, prime factor decomposition, fundamental theorem of arithmetic, highest common factor, HCF