So far, we have been introduced to the concept of factorising a given number. Factorising essentially means breaking down a number into a product of its divisors (numbers that do not give a remainder upon dividing). For example, the factors of the number 2 are 1 and 2. On the other hand, 4 has factors 1, 2, and 4. Now, notice how both 2 and 4 have two identical factors: 1 and 2. Could this be a coincidence?
Factors that appear twice upon factoring two numbers are known as common factors. This means that 1 and 2 are common factors of 2 and 4. In this lesson, we shall discuss the concept of a common factor and present a method we can use to identify common factors for a given pair of numbers.
Common factor: Definition
We shall begin this lesson by establishing the meaning of a common factor. Say we are given a pair of whole numbers, x and y. If we can divide these two numbers by a similar digit, then these numbers share a common divisor. This common divisor is known as the common factor of x and y.
Now, let us recall the definition of a factor.
Factors are numbers that can divide another number exactly resulting in a remainder of zero.
Linking this to our current study, we can say that a common factor is a number that divides two (or more) whole numbers precisely without leaving a remainder. We can summarise this definition as follows.
For a given pair of whole numbers (or more), a common factor is a factor shared by both (or all) these numbers.
Simple, right? A common factor must follow two specific rules or conditions: (1) it must be a factor and (2) it must be shared by the two (or all of the) numbers we're dealing with.
Let us continue this discussion by listing several properties of common factors.
Characteristics of common factors
Some notable characteristics of common factors are listed below.
Two (or more) numbers can have more than one common factor.
A common factor divides two (or more) numbers completely without leaving any remainder.
The common factor of two (or more) numbers is always less than the given numbers or equal to one of the given numbers.
The number 1 is always a common factor between two (or more) numbers.
Every non-zero whole number is a factor of 0 since any non-zero whole number times 0 equals 0.
We can summarise the last point as the following rule:
For any whole number k, if k × 0 = 0 then, 0 ÷ k = 0.
For example, given that 2 x 0 = 0, we have 0 ÷ 2 = 0. Therefore, 2 and 0 are factors of 0.
In the next section, we shall learn how to determine common factors for a given pair of numbers and observe some worked examples that apply this method.
Common factors method
Identifying common factors of two (or more) numbers follows a straightforward two-step method. This is described below.
Step 1: Write down all the factors of the given numbers in separate rows.
Step 2: Compare these lists of factors. Identify the recurring numbers in both lists and note these common factors.
Examples of common factors
Below are several worked examples for finding common factors.
Common factors between two numbers
Find the common factors between 15 and 25.
Solution
Now, following the two-step method we have:
Step 1: List the factors of the given numbers.
Factors of 15: 1, 3, 5, 15
Factors of 25: 1, 5, 25
Step 2: Check for repeating factors.
Looking at the lists above, we see that factors 1 and 5 are present in both lists.
Thus, the common factors of 15 and 25 are 1 and 5.
Based on this result, do you see any of the characteristics of common factors can here? Firstly, notice that both 1 and 5 divide 15 and 25 completely. There is no remainder upon division. Further, observe that the common factors of 15 and 25 happen to be less than both these numbers (1, 5 < 15, 25). Finally, we see that 1 is indeed a common factor between these given numbers.
Common factors between three numbers
Determine the common factors between 12, 16 and 20.
Solution
Again, following the two step-method we have:
Step 1: List the factors of the given numbers.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 16: 1, 2, 4, 8, 16
Factors of 20: 1, 2, 4, 5, 10, 20
Step 2: Check for repeating factors.
Looking at the lists above, we see that factors 1, 2 and 4 are present in all three lists.
Thus, the common factors of 12, 16 and 20 are 1, 2 and 4.
Common factors between four numbers
List down the common factors between 26, 52, 78 and 104.
Solution
Step 1: List the factors of the given numbers.
Factors of 26: 1, 2, 13, 26
Factors of 52: 1, 2, 4, 13, 26, 52
Factors of 78: 1, 2, 3, 6, 13, 26, 39, 78
Factors of 104: 1, 2, 4, 8, 13, 26, 52, 104
Step 2: Check for repeating factors.
Looking at the lists above, we see that factors 1, 2, 13, and 26 are present in all four lists.
Thus, the common factors of 26, 52, 78 and 104 are 1, 2, 13 and 26.
Importance of common factors
So why are common factors so necessary? Identifying common factors is actually a step toward our next topic of interest: Highest Common Factors (HCF). The HCF is the largest common factor of two or more numbers. Looking at our worked examples above, can you point out the HCF for each problem? Give it a go!
Common factors can also help us identify prime factors for a given set of whole numbers. Prime factors are factors of a whole number that is also a prime. As we list out the factors of two (or more) numbers and point out their shared divisors, we can subsequently recognise which of these common factors are also prime. In doing so, we can also represent a whole number as a product of primes. This is called prime factorisation which is discussed in more detail in the topic Prime Factorisation.
Knowing that we can conduct prime factorisation on a given number, also means that we can determine the lowest common multiple (LCM) between two (or more) numbers. The LCM is the smallest number that is a multiple of two numbers (or more) numbers. We do so by multiplying the common prime factors shared between a set of given numbers. This is discussed clearly in the article: Lowest Common Multiple.
Do you see how practical finding common factors are? The result allows us to find three other relationships between a given set of numbers: the HCF, prime factorisation and the LCM. Isn't that pretty nifty?
Representing common factors as a Venn diagram
Let us end this topic with two final examples of common factors. However, let's change this up a little. Here, we shall look at finding common factors between two (or more) numbers in the form of a Venn diagram. Representing common factors this way may be helpful when dealing with more than two numbers.
Find the common factors between 39 and 147.
Solution
Let us first list down the factors of 39 and 147.
Factors of 39: 1, 3, 13, 39
Factors of 147: 1, 3, 7, 21, 49, 147
Now, we shall sketch these values in a Venn diagram and note the overlapping factors.
Venn diagrams
From here, we conclude that the common factors of 39 and 147 are 1 and 3.
Find the common factors between 42, 51 and 108.
Solution
Let us first list down the factors of 42, 51 and 108.
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
Factors of 51: 1, 3, 17, 51
Factors of 108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108
Now, we shall sketch these values in a Venn diagram and note the overlapping factors.
Example 2 - Overlapping factors in Venn diagrams
From here, we conclude that the common factors of 42, 51 and 108 are 1 and 3.
This Venn diagram also shows that the common factors between 42 and 51 are also 1 and 3 , while 42 and 108 are 1, 2, 3 and 6 and finally 51 and 108 are again 1 and 3.
Common Factors - Key takeaways
- A common factor is a number that divides a pair of numbers precisely without leaving a remainder.
- Two numbers can have more than one common factor.
- Common factors can help us identify the highest common factor.
- Finding a common factor between two numbers:
- Write down the factors of the given numbers.
- List down the common (repeating) factors.
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