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 |