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Factors of 28: Prime Factorization, Methods, Tree, and Examples

Factors of 28 are the numbers that produce zero as a remainder whenever 28 is divided from such numbers. When these numbers act as divisors, they also produce a whole number quotient. 

In this case, both the quotient and the divisor act as factors for that number, and together they form a factor pair. The factors of 28 can be determined through various methods such as the division method and the prime factorization method.

An easy way to determine the factors of 28 is to look for the half of 28. Since the half of 28 is 14 so the total factors of 28 would lie between the smallest factor, which is 1, and the half of that number, in this case, 14.

The number 28 is also an even composite number which indicates that the number 2 has to be a factor of 28.

\[ \frac{28}{2} = 14 \]

As a whole number quotient is produced when 28 is divided by 2, hence the number 2 is a factor of 28. 

In this article, we will take a look at various methods and techniques which are used for determining the factors of 28. So let’s get started. 

What Are the Factors of 28?

The factors of 28 are 1, 2, 4, 7, 14, and 28. All these numbers yield zero as a remainder whenever 28 is divided from them. They also form factor pairs with their respective whole number quotients.

So in total, there are 6 factors in existence for the number 28. Similarly, there are 6 negative factors as well for the number 28. 

How To Calculate the Factors of 28?

You can calculate the factors of 28 through two main methods – the division method and the prime factorization method. But before calculating these factors, you first need to determine the range between these factors lies.

The smallest factor for any number is 1 so the range of factors of 28 begins with 1. Since half of 28 is 14, so the factors of 28 will lie between 1 and 14. 

Another thing to note is that the smallest factor for any number is the number 1 and the greatest factor for any number is the number itself. So in the case of 28, the smallest factor is 1 and the greatest factor is 28.

All the possible factors of 28 will produce a whole number quotient, so let’s take a look at these factors through the division method.

Since 28 is an even number, so let’s first consider the division of 28 through 2. This division is given below:

\[ \frac{28}{2} = 14 \]

Since a whole number quotient is produced, so 2 is a factor of 28. The division of other possible factors of 28 is shown below:

\[ \frac{28}{4} = 7 \]

\[ \frac{28}{7} = 4 \]

\[ \frac{28}{14} = 2\]

\[ \frac{28}{28} =1\]

So, the list of factors 28 is given below:

Factors of 28 = 1, 2, 4, 7, 14, and 28

In the same way, these factors can also be negative numbers. The negative factors of 28 are given below:

Factors of 28 = -1, -2, -4, -7, -14, and -28

Factors of 28 by Prime Factorization

Prime factorization is the method through which the prime factors for any number are determined. The process of the prime factorization is the same as the division with the exception that the prime numbers act as divisors.

This process of division carries on until 1 is obtained in the end. One thing to note in the prime factorization is that the divisors are always prime numbers.

The process of prime factorization for the number 28 is shown below:

\[ 28 \div 2 = 14 \]

\[ 14 \div 2 = 7 \]

\[ 7 \div 7 = 1\]

Since the result is 1 so this indicates that the prime factorization of 28 has been successfully conducted. This division also indicates that the prime factors of 28 are 2 and 7.

This prime factorization can be mathematically denoted as:

\[ \text{Prime Factorization of 28} = 2^{2} \times 7 \]

The prime factorization of 28 is also shown in figure 1 given below:

Figure 1

Factor Tree of 28

The factor tree is a visual representation of the prime factorization of any number. The factor tree begins with the number itself and then extends out its branches into a prime number and a whole number quotient.

The method of division in the factor tree is the same as that of the prime factorization. The only difference is that instead of ending the division at 1, as in the case of the prime factorization, the factor tree ends at prime numbers.

In the case of 28, the factor tree begins with 28 and after the first division step, it produces 2 and 14 as the output on its respective branches. The number 14 then acts as the dividend and produces 2 and 7 as the end products.

Since both 2 and 7 are prime numbers, so the factor tree concludes at this step.

The factor tree for the number 28 is shown below in figure 2:

Figure 2

Factors of 28 in Pairs

As mentioned above, the factors of 28 can also exist in the form of pairs. The division of the number 28 with a factor results in zero as the remainder and a whole number quotient.

This factor, which acts as the divisor, then forms a factor pair with its respective whole number quotient.

A factors pair includes the numbers that when multiplied together produce the original number as the product. The following factors form factor pairs for the number 28:

\[ 2 \times 14 = 28 \]

\[ 4 \times 7 = 28 \]

\[ 1 \times 28 = 28\]

So, below is the list of the factor pairs of 28:

Factor Pairs of 28 = (2, 14), (7, 4), and (1, 28)

Similarly, negative factor pairs of 28 can also exist. The only condition for negative factor pairs is that both the numbers existing in the pair need to be negative so they can together yield a positive product.

The negative factor pairs of 28 are given below:

Factor Pairs of 28 = (-2, -14), (-7, -4), and (-1, -28)

Factors of 28 as Solved Examples

To further strengthen the concept of the factors of 28, given below are a few examples.

Example 1

Find out the product of the even factors of 28.

Solution

To find the product of the even factors of 28, let’s first list down all the factors of 28. The factors of 28 are given below:

Factors of 28 = 1, 2, 4, 7, 14, and 28

The even factors of 28 are those which are divisible by 2, so the even factors of 28 are given below:

Even factors of 28 = 2, 4, 14, 28

The product of these even factors is given below:

\[ Product = 2 \times 4 \times 14 \times 28 \] 

Product = 3136 

So, the product of even factors of 28 is 3136. 

Example 2

Find the average of all the factors of 28.

Solution

For determining the average of all the factors of 28, let’s first list down all the factors of 28.

The factors of 28 are given below:

Factors of 28 = 1, 2, 4, 7, 14, and 28

The formula for calculating the average is given below:

\[ Average = \frac{\text{Sum of all factors}}{\text{Total no. of factors}}\]

\[ Average = \frac{1+2+4+7+14+28}{6}\]

\[ Average = \frac{56}{6} \]

Average = 9.334

So the average of all the factors of 28 is 9.334.

Example 3

Find the sum of the common factors between 28 and 20.

Solution

To determine the sum of the common factors between 28 and 20, let’s first list these factors.

Factors of 28 = 1, 2, 4, 7, 14, and 28

Similarly, the factors of 20 are:

Factors of 20 = 1, 2, 4, 5, 10, 20

The common factors between any two numbers are the identification numbers that act as the factors for both numbers.

In this case, the common factors of 28 and 20 are given below:

Common factors = 1, 2, 4

The sum of these common factors is given as:

Sum = 1 + 2 + 4

Sum  = 7

So, the sum of the common factors between 28 and 20 is 7.

Example 4

Calculate the difference between the sum of odd factors and the even factors of 28.

Solution

For calculating the difference between the sum of odd factors and even factors of 28, let’s first list down the factors of 28.

Factors of 28 = 1, 2, 4, 7, 14, and 28

The odd factors of 28 are given below:

Odd factors of 28 = 1, 7

The even factors of 28 are given below:

Even factors of 28 = 2, 4, 14, 28

Now, let’s calculate their sum.

Sum of odd factors = 1 + 7

Sum of odd factors = 8 

Similarly,

Sum of even factors = 2 + 4 + 14 + 28

Sum of even factors = 48

The difference between the two sums is given as:

Difference = 48 – 8

Difference = 40

All images/mathematical drawings are created with GeoGebra. 

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