Factors

Cheryl wants to decorate one of the walls of her room with some pictures of her friends and family. She has 24 printed photos in total. She realises that she is not able to fit all 24 pictures horizontally in a single line. She then tries to make 2 lines of 12 photos each. Unfortunately, this did not fit her wall either. Next, she made 3 lines of 8 photos each to see if this arrangement would work instead. Still, the wall could not hold this many photos lengthwise. Finally, she laid out the pictures in 4 lines of 6 photos each. Now all the photos can fit the wall perfectly!

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Jetzt kostenlos anmeldenCheryl wants to decorate one of the walls of her room with some pictures of her friends and family. She has 24 printed photos in total. She realises that she is not able to fit all 24 pictures horizontally in a single line. She then tries to make 2 lines of 12 photos each. Unfortunately, this did not fit her wall either. Next, she made 3 lines of 8 photos each to see if this arrangement would work instead. Still, the wall could not hold this many photos lengthwise. Finally, she laid out the pictures in 4 lines of 6 photos each. Now all the photos can fit the wall perfectly!

Example 1: Factors

Notice how there were four ways in which Cheryl could organise these photos on her wall. The first way could be 1 x 24, the second could be 2 x 12, the third could be 3 x 8 and finally, the one that worked: 4 x 6. So, what do these products of numbers entail? This is where our topic comes in! The numbers 1, 24, 2, 12, 3, 8, 4 and 6 are called **factors **of 24.

Let's discuss the concept of a factor and identify factors of a given number. We will begin with the definition of a factor.

A factor is essentially a **divisor** of a given number. Dividing a number by its divisor results in a remainder equal to zero. As we have seen, the number 24 has a total of 8 factors, namely, 1, 2, 3, 4, 6, 8, 12 and 24.

Now, try this yourself: divide 24 by each of these listed factors. You will notice that upon dividing them, there will be no remainders!

Here is a question: can factors be negative numbers? Algebraically, we can indeed divide a number by one that is negative. Let us look at the factors of the number 6. We can factorise 6 in the following 2 ways: 1 × 6 and 2 × 3. Therefore, the factors of 6 are 1, 2, 3 and 6.

We know that the product of two negative numbers yields a positive number. Taking this into context, we find that the factors of 6 are in fact, 1, –1, 2, –2, 3, –3, 6 and –6 since –1 × (–6) and –2 × (–3) also equate to 6. However, this article will not take into account negative factors. When dealing with problems related to factors, we shall only consider the positive factors.

There are several important properties of factors that we should familiarise ourselves with. These are listed below.

There is a finite number of factors for a given number.

The factor of a given number is always less than or equal to itself.

Every number has at least two factors, that is, 1 and itself (except 0 and 1).

We can use division and multiplication to find factors.

What are the factors of 18?

**Solution**

Let us first write down the pair of divisors in which their product yields 18:

1 × 18 = 18

2 × 9 = 18

3 × 6 = 18

There are 6 factors to the number 18, namely 1, 2, 3, 4, 9, and 18.

To find the factors of 18 in this example, we used the method of multiplication. However, there is another way for us to tackle this problem. The next section shall cover this!

Looking at the tip in Characteristics of Factors, there are two methods we can use to find factors: **multiplication **and **division**. Let us observe both techniques in turn.

There are two steps to finding factors using multiplication.

Express the given whole number as a product of two whole numbers. You must consider every possible way in which this number can be written in this manner;

List down all the numbers that are involved in these products. These are the factors of the given number.

Use multiplication to find the factors of 27.

**Solution **

We need to look for all the possible ways in which we can write 27 as a product of two numbers. In doing so, we obtain the following combinations.

1 × 27 = 27

3 × 9 = 27

All the numbers seen in these products of 27 above are its factors. Thus, the factors of 27 are 1, 3, 9, and 27.

As for division, this follows a three-step method.

Identify all the whole numbers less than or equal to the given whole number.

Divide the given number by each of the numbers found in Step 1.

Write down the divisors that give the remainder of 0. These are the factors of the given number.

Use division to find the factors of 5.

**Solution **

Let us begin by listing down all the positive numbers that are less than or equal to 5. These are 1, 2, 3, 4, and 5. Now, we shall divide 5 by each of these numbers one by one.

Dividing by 1: $155-5\mathbf{0}$ Dividing by 2: $225-41$ Dividing by 3: $315-32$Dividing by 4: $415-41$ Dividing by 5: $515-5\mathbf{0}$

Here, we see that the number 1 and 2 are divisors of 5 since the remainder equals zero in this case. Hence, the factors of 5 are 1 and 5.

Applying division when determining factors can be somewhat lengthy at times, especially when dealing with larger numbers. Always be extra cautious of the algebra involved here and be sure not to miss any numbers upon dividing! However, in any case, it would indeed be safer to use the multiplication method to find the factors of a given number.

Sometimes, it may be helpful to know the number of factors a number may have. This is particularly useful, especially when dealing with larger numbers. To do this, we need to be made aware of a new concept called prime factorisation. Taking this idea into account, we can identify the number of factors for a given number using a process called the factor tree method.

So what do prime numbers and factors have in common? Firstly, let us recall what a prime number is. A prime number is a number that has precisely two factors, 1 and the number itself. For example, the factors of 15 are 1, 3, 5, and 15. Based on this list of factors, we find that 3 and 5 are prime numbers. This brings us to the following definition.

A **prime factor** is a factor of a given number that is also a prime.

So far, we have seen how we can represent a number as a product of two numbers. However, we can also express a number as a product of its prime factors. This is called prime factorisation.

**Prime factorisation** is the process of writing a number as a product of its prime factors.

Essentially, what we are doing here is breaking down a number in terms of its prime factors. There are two ways to determine the prime factorisation of a number.

- Factor tree method
- Division method

In this context, we shall only focus on the first method as it is widely used to find the number of factors for a given number. A more detailed explanation discussing this subtopic can be found in our explanation on Prime Factorisation.

We shall first establish the factor tree method below.

Write the number at the top of the factor tree.

Express the number as a product of two factors branching out of the tree.

Further branch out each of these factors found in Step 2 as a product of two factors.

Repeat Step 3 until we are unable to branch out each factor. At this point, it should be written as a prime factor.

Finally, define the given number as a composite of its prime factors in exponent form.

Here is an example that demonstrates this process.

Use the factor three method to factorise the number 132 in terms of its prime factors.

**Solution **

The factor tree for 132 is displayed below.

Example 2: Factor Tree of 132

Based on this factor tree, we can write 132 as the following.

132 = 2 × 2 × 3 × 11

In exponent form, we have

132 = 2^{2} × 3 × 11

From here, we can identify the number of factors using the factor tree method. There are four steps to this method.

Find the prime factorisation of the given number using the factor tree method.

Express this found product of primes in its corresponding exponent form.

Add 1 to each exponent.

Multiply the numbers found in Step 3. The result yields the number of factors of the given number.

To show this, let us use our previous example.

Find the number of factors for the number 132.

**Solution **

By the factor tree method conducted previously, we can express 132 as 132 = 2 × 2 × 3 × 11.

In exponent form, we obtain 132 = 2^{2} × 3 × 11 (2^{2} × 3^{1} × 11^{1} ).

Here, we have the following values of exponents.

Exponent for 2 = 2

Exponent for 3 = 1

Exponent for 11 = 1

Now, adding 1 into each of these exponents yields

Exponent for 2 **+ 1** = 3

Exponent for 3 **+ 1** = 2

Exponent for 11** + 1** = 2

Multiplying these numbers, we obtain

3 × 2 × 2 = 12

Thus, the number 132 has 12 factors.

**Check**

Let us verify if our result is correct. Using the multiplication method, we may write 132 as the following products of two numbers.

1 × 132 = 132

2 × 66 = 132

3 × 44 = 132

4 × 33 = 132

6 × 22 = 132

11 × 12 = 132

The factors of 132 are 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66 and 132. Thus, the number 132 has a total of 12 factors, as stated.

Let's say we are told to compare the factors between two whole numbers x and y. We find that they share a divisor that indeed divides both numbers without leaving a remainder. This is called the **common factor** between x and y.

A **common factor** is a number that divides a pair of whole numbers precisely without leaving a remainder.

To find the common factor between two (or more) numbers, we simply list down the factors of each number in separate rows and check for overlapping factors. A factor that appears in both (or more) lists is the common factor. Here are two worked examples that demonstrate this.

Find the common factor(s) between 14 and 21.

**Solution **

Let us first identify the factors of 14. Using the multiplication method, we may write 14 as the following products of two numbers.

1 × 14 = 14

2 × 7 = 14

Doing the same for 21, we obtain

1 × 21 = 21

3 × 7 = 21

Listing these down, we have

Factors of 14: 1, 2, 7, 14

Factors of 21: 1, 3, 7, 21

Looking at the lists above, we see that factors 1 and 7 are present in both lists. Thus, the common factors of 14 and 21 are 1 and 7.

Find the common factor(s) between 4, 12, and 16.

**Solution **

We shall first look for the factors of 4. Using the multiplication method, we may express 4 as the following products of two numbers.

1 × 4 = 4

2 × 2 = 4

Doing the same for 12, we obtain

1 × 12 = 12

2 × 6 = 12

3 × 4 = 12

Writing this for 16 as well yields

1 × 16 = 16

2 × 8 = 16

4 × 4 = 16

Listing these down, we have

Factors of 4: 1, 2, 4

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 16: 1, 2, 4, 8, 16

Looking at the lists above, we see that factors 1, 2, and 4 are present in all three lists. Thus, the common factors of 4, 12, and 16 are 1, 2, and 4.

A more detailed discussion on this topic is presented in the article: Common Factors.

In this section, we will be introduced to a direct consequence of common factors called the highest common factor (HCF). The HCF is the largest common factor of two or more whole numbers. Let us denote the HCF by the letter a. Thus, we can write the HCF of x and y by HCF(x, y) = a.

The **highest common factor** (HCF) of two numbers is the largest possible number that divides both x and y without leaving a remainder.

There are three ways to find the HCF of two (or more) numbers.

- Listing factors method
- Prime factorisation
- Division method

Among these three techniques above, the listing factors method is the most straightforward of them all. Here, we simply list the factors of each number and find the common factors of those numbers. After that, we determine the highest common factor among this list of common factors.

For a more thorough explanation of this topic explaining all these methods, you can refer to Highest Common Factor. To show this, we shall return to the last two examples from our previous section.

What is the HCF of 14 and 21?

**Solution**

We shall first list the factors of 14 and 21.

Factors of 14: 1, 2, 7, 14

Factors of 21: 1, 3, 7, 21

From this list, we find that the common factors of 14 and 21 are 1 and 7.

Here, the highest common factor between 14 and 21 is 7.

Thus, HCF(14, 21) = 7

What is the HCF of 4, 12, and 16?

**Solution**

We shall first list the factors of 4, 12, and 16.

Factors of 4: 1, 2, 4

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 16: 1, 2, 4, 8, 16

From this list, we find that the common factors of 4, 12, and 16 are 1, 2, and 4.

Here, the highest common factor between 4, 12, and 16 is 4.

Thus, HCF(4, 12, 16) = 4

Finding factors of numbers is a very important tool in mathematics. It is predominantly used in mathematics to solve ratios and fractions. Factors also play a significant role in algebra, especially when dealing with equations. In this case, we can use this skill to recognise patterns found in given expressions, and reduce (or expand) such equations as a means of solving them.

As mentioned before, we can use factors in algebra when working with polynomials. For example, say we are given the expression 4x. The factors of 4x are 1, 2, 4, x, 2x, and 4x. As we shall uncover in later topics, we can use factors to perform algebraic calculations such as factoring expressions and polynomials. Here are a few more examples that demonstrate this.

Find the factors of 3x^{2}.

**Solution**

The factors of 3x^{2} are 1, 3, x, 3x, x^{2 }and 3x^{2}

Find the factors of 5xy.

**Solution **

The factors of 5xy are 1, 5, x, 5x, y and 5y

In real-life situations, factoring numbers is mostly used when arranging is involved, as you have seen at the beginning of this topic. We can use factors to divide an object (or a set of items) into equal parts or split them into rows and columns.

We shall end this topic by looking at the difference between factors and multiples. We will do this by means of the following table.

Factors | Multiples |

A factor of a number is a number that divides the original number completely without any remainder. | A multiple of a number is a product of that number and any other number. |

There are a finite number of factors for a given number. | There are an infinite number of multiples for a given number. |

The factors of a number are always less than or equal to the initial number. | The multiples of a number are always more than or equal to the initial number. |

Every number has at least two factors: 1 and the number itself. | There is no minimum number of multiples for a given number. |

- A factor of a given number is one that divides the number without leaving a remainder
- We shall only consider positive factors of a given number
- There is a finite number of factors for a given number
- The factor of a given number is always less than or equal to itself
- Every number has at least two factors, that is, 1 and itself
- We can use division and multiplication to find factors
- Prime factorisation is the process of writing a number as a product of prime numbers
- A common factor is a number that divides a pair of numbers precisely without leaving a remainder
- The highest common factor (HCF) of two numbers is the largest possible number that divides both x and y without leaving a remainder
- We can use factors to factorise algebraic expressions and polynomials

The factors of a number are numbers that divide a number without leaving a remainder

A prime factorization is a factorization of a number in which every factor is a prime number

We can use the greatest common factor (GCF) to factor quadratics

The factors of 20 are, 1, 2, 4, 5, 10, 20

What is a common factor?

A common factor is a number that divides a pair of numbers precisely without leaving a remainder

What are the common factors of 8, 12, 20 and 28?

1, 2 and 4

What are the common factors of 68 and 88?

1, 2 and 4

What are the common factors of 16 and 18?

1 and 2

What are the common factors of 15 and 21?

1 and 3

What are the common factors of 12 and 16?

1, 2 and 4

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