**RECAP **

Factors are a pair or group of numbers that are multiplied to obtain a product.

**Definition of Factors of 40**

Using the definition above, factors of 40 are numbers used to multiply to get the product of the number itself.

There are various methods to get the factors of a number. For this case, we will discuss the different ways to get the factors of 40.

**Methods of Getting the Positive Integer Pair of Factors of 40**

**Method 1: Prime Factorization Using Tabular Division**

First, we are going to make a table with two columns.

Next, put 40 to the left of the table.

40 |

Then, write the smallest prime factor that can divide 40. Since 40 is an even number, we can use 2 as its first prime factor.

2 | 40 |

At this point, we can divide 40 by 2. But in order to have a space for the quotient, we need to add another row. After that, put the quotient below 40.

2 | 40 |

20 |

After obtaining the quotient, repeat the process until reaching the quotient of 1.

2 | 40 |

2 | 20 |

2 | 10 |

5 | 5 |

1 |

Now that we reached the quotient of 1, we can end the process.

To get the first pair of factors of 40, by default, we can declare 40 and 1 as its first factors.

2 | 40 |

2 | 20 |

2 | 10 |

5 | 5 |

1 |

40=1×40

Then, the second pair of factors will be the first divisor and quotient.

2 | 40 |

2 | 20 |

2 | 10 |

5 | 5 |

1 |

40=2×20

The third pair of factors will be the first two divisors and the second quotient.

2 | 40 |

2 | 20 |

2 | 10 |

5 | 5 |

1 |

40=2×2×10

40=4×10

And the last pair will be the first three divisors and the second to the last quotient.

2 | 40 |

2 | 20 |

2 | 10 |

5 | 5 |

1 |

40=2×2×2×5

40=8×5

Overall, the factors of 40 are the following:

40=1, 40, 2, 20,4, 10, and (8, 5)

Moreover, we found out that there are 4 pairs (8 in total) of positive integer factors to obtain a product of 40.

**Method 2: Prime Factorization Using Factor Tree**

To use this method, we need to split the number until reaching its prime factor or to the point that we cannot split factors anymore. The only similarity from the previous method is that we can start with a pair that has its smallest prime factor. In the case of 40, we can use 2 and its partner factor (i.e. 20) again to start the Factor Tree.

Again, we can consider already 40 and 1 as the first factors of 40 even it is not written in the factor tree. (Technically we can break down 5 and 2 by splitting them into their factors which are 1×2 and 1×5 respectively. However, 1 is not a prime factor, so we cannot include it in our Prime Factorization.)

The second level of the tree will be the 40’s second pair of factors.

40=2×20

Then, the third level of the tree will be the third pair of factors. But, we need to include the first prime factor.

40=2×2×10

40=4×10

For the final pair of factors, we need to include the ground level of the tree and the first two prime factors.

40=2×2×2×5

40=8×5

Listing all the factors of 40 using the Factor Tree Method, we have the following:

40=(1, 40), (2, 20), (4, 10), and (8, 5)

Similar to the prior method, we also obtained 4 pairs (8 in total) of positive integer factors of 40.

**Method 3: Listing Method**

Now that we already have the idea of what are the factors of 40, we can now memorize them by heart. But to have a formal convention of listing the factors as elements, we need to list them from least to greatest.

*Step 1:* Start with 1 and 40, writing 40 as the last element since it is the highest among them all.

40={1, 40}

* Step 2:* Then, write the next factors as the inner parts of the set.

2 and 20

40={1, 2, 20, 40}

4 and 10

40={1, 2, 4, 10, 20, 40}

5 and 8

40={1, 2, 4, 5, 8, 10, 20, 40}

**Method 4: Using Divisibility Rules**

*The Divisibility Rule states, “A number is divisible by another number if and only if the first number divides evenly into the second number, without remainder.”*

With this rule, we can quickly find the factors of 40 without using the previous rules. But please take note that every number is divisible by 1. This means that we can include 1 and 40 as factors of 40 itself by default.

**2** – a number is divisible by 2 if it ends with an even number (e.g. 0, 2, 4, 6, 8)

**3** – if the sum of the digits can be divided evenly by 3

Example: 2070, because 2 + 0 + 7 + 0 = 9, wherein 9 is divisible by 3.

**4** – if the last two digits can be divided exactly by 4

Example: 7800, 6328, and 1512

**5** – if the number ends with 0 or 5

Example: 2500, 5655, and 34565

**6 **– if the number is divisible by both 2 and 3

Example: 132, because it is an even number; also, 1 + 3 + 2 = 6

**7 **– if you multiply the last digit by 2 and subtract the product to the remaining digits and you get a number that can also be divided by 7 evenly

Example: 49, 9 x 2 = 18

4 – 18 = -14 , where -14 is divisible by 7

**8** – if the last three digits of the number are also divisible by 8

Example: 89024, 7816, and 9800

**9** – if the sum of the digits is divisible by 9

Example: 1305, since 1 + 3 + 0 + 5 = 9, where 9 is divisible by 9

**10** – if the number ends with 0

Example: 2020, 1960, and 1800

**Following the rules above, let us have a checklist for each number to verify if 40 is**** ****divisible by each one of them.**

Number | Is 40 divisible by the number? | Why? |

2 | Yes | 40 is an even number. |

3 | No | 4 + 0 = 4, which cannot be divided exactly by 3. |

4 | Yes | 40 can be divided fairly by 4, having an exact quotient of 10. |

5 | Yes | 40 ends with 0. |

6 | No | 40 is divisible by 2, but not by 3 because 4 + 0 = 4. |

7 | No | 0, which is the last digit of 40 cannot be divided exactly by 7. |

8 | Yes | Even though 40 is a two-digit number, we can still divide it by 8 evenly and get 5 as an exact quotient. |

9 | No | The sum of the digits of 40 is 4, not 9. |

10 | Yes | 40 ends with 0. |

Looking at the data above, we can easily tell 40 is divisible by 2, 4, 5, 8, and 10. After discovering the numbers that 40 are divisible by, we can divide each number by 40 to obtain the pair of factors of 40.

40÷2=20

So, 2 and 20 are factors of 40.

40÷4=10

Hence, 4 and 10 are factors of 40.

40÷5=8

Thus, 5 and 8 are factors of 40 as well.

Since we already got 8 and 10 earlier, we can now end the process since Multiplication has a commutative property anyway. If a mathematical operation is commutative, this means we can get the same answer even we change the order of the numbers.

For instance, we can get 40 as a product even we multiplied 5 by 8 and vice versa.

40=5×8

40=8×5

The same thing goes for 4 and 10 as factors of 40.

40=4×10

40=10×4

With this method, we can also conclude that 40 has 4 pairs of factors (1, 40), (2, 20), (4, 10), and (5, 8).

**In a nutshell, we have 8 positive integer factors of 40: 1, 2, 4, 5, 8, 10, 20, and 40.**

**Proper Factors of 40**

Proper Factors are factors that do not include the number itself as a factor. With that in mind, we cannot include 40 as a proper factor in itself.

**Thus, we only have 7 proper factors 40: 1, 2, 4, 5, 8, 10, and 20.**

**Sum of Factors of 40**

To sum up all the positive integer factors of 40, we have

1+2+4+5+8+10+20+40=90

However, if we are considering only the proper factors of 40, we only have

1+2+4+5+8+10+20=50

**40 as an Excessive Number**

A number is excessive (also known as Abundant Number) if the sum of its proper factors is greater than the number itself. Earlier, we discovered that the sum of proper factors of 40 is 50. Since 50 is greater than 40, we can declare 40 as an Excessive Number. This implies that 40, as well as other excessive numbers, has an excess amount when being made especially in real-life situations.

**Prime Factors of 40**

Prime numbers are numbers that have factors of 1 the number itself. To know what the prime factors of 40 are, let us go back with the prime factorization of 40.

Using the Factor Tree, we can see that 40 has prime factors of

40=2×2×2×5

Or if we are going to shorten the convention, we have

40=2^{3}×5

**Hence, the prime factors of 40 are 2, 2, 2, and 5 or 2**^{3}** and 5.**

**Negative Integer Factors of 40**

By definition, we can also have factors of 40 that are negative integers. And we can derive those by following the rule of multiplying negative integers. According to the rule, we can only obtain a positive product if the factors are either negative or positive.

+×+ =+

-×- =+

Since 40 is a positive integer, this means we need to transform the factors we got earlier from positive integers into negative integers.

40=1×40 | 40=-1×-40 | |

40=2×20 | 40=-2×-20 | |

40=4×10 | 40=-4×-10 | |

40=5×8 | 40=-5×-8 |

Thus, the negative factors of 40 are –**1, -2, -4, -5, -8, -10, -20, and -40.**

**Rational Factors of 40**

Rational numbers not only include integers. This set of numbers also includes fractions and decimals. With this in mind, we can also have factors of 40 that are in form of fractions and decimals. But unlike integers, fractions and decimals are not exact numbers. As a result, we can get infinite fractional and decimal factors of 40.

**Fractional Factors of 40**

We can get infinite factors of 40 that one or both of them are fractions. Using Tabular Division, we can see infinite pairs of factors of 40 if we use the reciprocal of its smallest prime factor (2) as the starting divisor.

2 reciprocal $\frac{1}{2}$

So, let us start with 12 as the first divisor.

$\frac{1}{2}$ | 40 |

80 |

40 ÷ $\frac{1}{2}$ → 40 × $\frac{1}{2}$ → 80

40=$\frac{1}{2}$ ×80

Next, divide 80 by $\frac{1}{2}$ again.

$\frac{1}{2}$ | 40 |

$\frac{1}{2}$ | 80 |

160 |

40=$\frac{1}{2}$×$\frac{1}{2}$×160

40=$\frac{1}{4}$×160

Then, divide 160 by $\frac{1}{2}$ as well.

$\frac{1}{2}$ | 40 |

$\frac{1}{2}$ | 80 |

$\frac{1}{2}$ | 160 |

… | 320 |

40=$\frac{1}{2} × \frac{1}{2} x \frac{1}{2}$×160

40=$\frac{1}{8}$×32

At this point, we can notice that the pattern is never-ending. The reason for this is having a fraction, which is not an exact number, as the divisor.

So the first factors of 40 with fractions as one of the pairs are $\frac{1}{2}$, $\frac{1}{4}$,$\frac{1}{8}$, 80, 160, and 320.

Furthermore, we can have 40 also as one of the pair of factors. Let us have an illustration of when 40 is divided into 8 parts.

40

5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 |

↑$\frac{1}{8}$ | ↑$\frac{1}{4}$ | ↑$\frac{3}{8}$ | ↑$\frac{1}{2}$ | ↑$\frac{5}{8}$ | ↑$\frac{3}{4}$ | ↑$\frac{7}{8}$ | ↑1 |

Using the illustration above, we can get a list of fractional factors of 40 below and itself. But be careful not to declare 5 and $\frac{1}{8}$ as factors because we cannot get a product of 40 by multiplying those. Instead, we are going to get the reciprocal of the fractions to set as factors.

Getting the first pair of factors:

$\frac{1}{8}$ reciprocal 8

40=5 ×8

40={5, 8}

The second pair of factors:

$\frac{1}{4}$ reciprocal 4

40=10 ×4

40={10, 4}

Third pair of factors:

$\frac{3}{8}$ reciprocal $\frac{8}{3}$

40=15 ×$\frac{8}{3}$

40={15, $\frac{8}{3}$}

Fourth pair of factors:

$\frac{1}{2}$ reciprocal 2

40=20 ×2

40={20, 2}

Fifth pair of factors:

$\frac{5}{8}$ reciprocal $\frac{8}{5}$

40=25 ×$\frac{8}{5}$

40={25, $\frac{8}{5}$}

Sixth pair of factors:

$\frac{3}{4}$ reciprocal $\frac{4}{3}$

40=30 × $\frac{4}{3}$

40={30, $\frac{4}{3}$}

And the last pair of factors:

$\frac{7}{8}$ reciprocal $\frac{8}{7}$

40=35 ×$\frac{8}{7}$

40={35, $\frac{8}{7}$}

Scraping the data above, we can have fractional factors of 40 are as follows:

40={15, $\frac{8}{3}$}, {25, $\frac{8}{5}$}, and {35, $\frac{8}{7}$}

If we want to have factors of 40 that are both fractions, we can divide the first number in each pair by 2 and multiply their partners by 2.

Overall, the basic pairs of factors of 40 that are in form of fractions are

{$\frac{15}{2}, \frac{16}{3}$}, {$\frac{25}{2}$, $\frac{16}{5}$}, {$\frac{35}{2}, \frac{16}{7}$}, {15, $\frac{8}{3}$},** **{25, $\frac{8}{5}$}, {35, $\frac{8}{7}$}, {80, $\frac{1}{2}$}, {160, $\frac{1}{4}$}, and {320, $\frac{1}{8}$}.

**Decimal Factors of 40**

Since fractions can be expressed into decimals. We can also have factors 40 that are in decimal form. All we can do is transform the fractions into decimal forms.

For example, we can transform the following fractions into decimals.

{152, 163} -> {7.5, 5.$\overline{3}$}

{252, 165} -> {12.5, 3.2}

{352, 167} -> {17.5, 2.28…}

{15, 83} -> {15, 2.$\overline{6}$}

{25, 83} -> {25, 1.6}

{35, 87} -> {35, 1.14…}

{80, 12} -> {80, 0.5}

{160, 14} -> {160, 0.25}

{320, 18} -> {320, 0.125}

Thus, the basic factors of 40 in decimals are 0.125, 0.25, 0.5, 1.14…, 1.6, 2.28…, 2.$\overline{6}$, 3.2, 5.$\overline{3}$, 7.5, 12.5, 15, 17.5 25, 35, 80, 160, and 320.

**Group Factors of 40 **

Often, the factors of a number come in pairs. However, if we go back to the definition of factors, we can also have more than two numbers to multiply and get a certain product.

For instance, if we go back to the prime factorization of 40, we have

2×2×2×5=40

Here, we have 4 numbers to multiply to obtain a product of 40.

Let us have some brain teasers to guess what factor will complete the following group of factors of 40.

**Example 1:**

? ×2×5=40

*Solution: *

First, multiply 2 by 5.

? ×2×5=40

? ×10=40

Then, divide 40 by 10.

? ×10=40

40÷10=4

Since 4 is the quotient, this means that 4 is the missing factor in this group of factors of 40.

? ×2×5=40

4×2×5=40

Thus, we can obtain a product of 40 having the factors **2, 4, and 5**.

**Example 2:**

? ×-4×10=40

*Solution: *

Again, we can multiply the given factors first.

? ×-4×10=40

? ×-40=40

Then, divide the 40 by -40.

? ×-40=40

40÷-40=-1

This means that -1 is the missing product for this group of factors of 40. Moreover, we can get a product of 40 by multiplying the factors -1, -4, and 10.

**Example 3:**

? ×8×10=40

*Solution:*

First, let us try to multiply the given factors.* *

? ×8×10=40

? ×80=40

Then, divide 40 by 80. In this case, our dividend (40) is lower than our divisor (80). So, we are just going to reduce them to the lowest terms.

? ×80=40

40÷80 -> $\frac{40}{80}$ -> $\frac{1}{2}$

Thus, the missing factor in this group is $\frac{1}{2}$. This means we can get also a product of 40 having the factors of $\frac{1}{2}$, 8, and 10.

Having those examples, we can say that we can also have factors of 40 that are not in pairs.

**Factors of 40 VS Multiples of 40**

Factors and multiples are commonly interchanged, which is certainly incorrect to do. Factors are numbers used to multiply, while multiples are a series of products from one certain number.

Factors of 40 include 1, 2, 4, 5, and so on because these numbers will “make up” 40. While multiples of 40 include 40, 80, 120, 160, and so on because these are the numbers that can be obtained when 40 is being multiplied starting from 1 until infinity.

** factors of 40** ↓ 1×40 2×20 =40

x y =40

4×10 =40 2×40 =80

5×8 =40 3×40 =120

x y 1×40 ** multiples of 40** ↓ =40 4×20 =160

5×20 =20

**Factors of 40 in Real-Life Situations**

The principles of factors are widely used in our daily situations. The following instances will show how we can use the factors of 40 to solve real-life situations.

- Finding the original price of an item

Elsa is in a grocery store and found out that the 10 kilograms of rice cost $40, which is already discounted by 20%. She is curious about how much is the original price of the rice So, what is the original price of the 10 kilograms of rice?

Solution:

If 20% is being discounted, this means that 80% comprises the price of $40.

Or in equations,

Original Price 80% = $40

So, we need to find the other factor of 40 by dividing it by 80% or 0.8 in decimal form.

40 ÷ 0.8 = 50

The pair factor, in this case, is 50 and 0.8. Thus, $50 is the original price of the 10 kilograms of rice that has been discounted by 20%.

- Considering it as a Large Quantity

Did you remember that 40 is an excessive or abundant number? The excessiveness of a number conveys that it is more than its worth. And we knew that 40 is an excessive number by summing up all of its positive integer factors.

**Key Facts and Summary**

- Factors of 40 are pairs or groups of numbers that are multiplied to get a product of 40.
- There are different techniques of getting the factors of 40:
- Prime Factorization using Tabular Division
- Prime Factorization using a Factor Tree
- Listing Method
- Usage of Divisibility Rules

- There are 8 positive integer factors of 40: 1, 2, 4, 5, 8, 10, 20, and 40. But considering only the proper factors of 40, we only have seven: 1, 2, 4, 5, 8, 10, and 20.
- The sum of all pairs of factors of 40 is 90. While the sum of all its proper factors is 50, which makes 40 an Excessive number.
- The prime factors of 40 are 2
^{3}and 5. - There are also 8 negative integer factors of 40: -1, -2, -4, -5, -8, -10, -20, and -40.
- We can also have rational factors of 40 that are in the form of fractions or decimals.
- Factors of 40 are numbers needed to multiply to get a product of 40. While multiples of 40, are a series of products from multiplying 40 by 1 until infinity.
- We can use finding the factors of 40 in real-life situations.

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