Factors and the Prime Factorization of 84

In mathematics, factors are crucial. They describe the becoming of a number, and what it can become when multiplied with another number. Factors of a number show how a number becomes what it is when mutliplying. Some numbers only have 2 factors (prime numbers), while others can have many. We will now discuss the factors of 84. Before doing that, we’ll understand what factors are and how to find them.

What are Factors?

When we multiply numbers, they create a product. Thus, the numbers that created that product are called its factors. Numbers can have 1 to n numbers as its factor. For example, for 1, the only factor is 1, but for another such as 6, it has 2 factors, i.e., 2 and 3:

1 = 1 x 1 therefore, the only factor of 1 is 1.

6 = 2 x 3 therefore, the 2 factors of 6 are 2 and 3.

What are Prime Numbers?

Prime numbers are numbers that only have 2 or less factors, which are the number itself and the number 1. For example, the numbers 1, 2, 3, 5, 7, etc. All of these numbers have only 2 factors, as is depicted below.

1 = 1 x 1 thus the only factor is 1
2 = 1 x 2 thus the only factors are 1 and 2
3 = 1 x 3 thus the only factors are 1 and 3
5 = 1 x 5 thus the only factors are 1 and 5
7 = 1 x 7 thus the only factors are 1 and 7

How to Find Factors of a Number?

When you are looking for the factors, you have to multiply numbers which are less than it to find that specific number. For example, here are the factors of 20 found by multiplying:

1 x 20 = 20
2 x 10 = 20
4 x 5 = 20

Therefore, the factor pairs of 20 are (1, 20), (2, 10), (4,5), and the factors of 20 are 1, 2, 4, 5, 10, and 20.

What is Prime Factorization?

Prime factorization refers to dividing numbers with prime numbers successively to find its prime factors. Start with the smallest prime number, then move upwards until you have a whole number. Finish your prime factorization when 1 is the quotient. For example: 20.

Divide by smallest prime factor: 20 ÷ 2 = 10

Divide quotient 10 by the smallest prime factor again: 10 ÷ 2 = 5

Now, 5 ÷ 2 = 2.5, 5 ÷ 3 = 1.67, and 5 ÷ 5 = 1, thus we take the last option.

The prime factorization for 20 is, hence, 5 x 2 x 2 = 20. It can also be written as 20 = 2² x 5

What Are the Factors of 84?

You can find factors of number 84 by multiplying numbers to obtain a product of 84.

1 x 84 = 84
2 x 42 = 84
3 x 28 = 84
4 x 21 = 84
6 x 14 = 84
7 x 12 = 84

Factor pairs of 84:
(1, 84)
(2, 42)
(3, 28)
(4, 21)
(6, 14)
(7, 12)

Factors of 84: 1, 2, 3 , 4 , 6, 7, 12, 14, 21, 28, 42, 84

Prime Factorization of 84

To do the prime factorization of 84, successively divide 84 by prime numbers.

Step 1: Divide 84 by smallest prime number: 2

84 ÷ 2 = 42

Step 2: Divide quotient 42 by smallest prime number

42 ÷ 2 = 21

Step 3: Divide quotient 21 by smallest prime number

21 ÷ 2 = 10.5

10.5 is a decimal, not a whole number, so we can’t take this.
Divide 21 by the next smallest prime number: 3

21 ÷ 3 = 7

Step 4: Divide quotient 7 by smallest prime number (or next smallest prime number until you reach a whole number)

7 ÷ 2 = 3.5
7 ÷ 3 = 2.33
7 ÷ 5 = 1.4
7 ÷ 7 = 1

Now we have reached quotient 1, we can’t divide any further. Thus our prime factorization of 84 is

84 = 2 x 2 x 3 x 7 = 2² x 3 x 7

Factor Tree of 84

A factor tree of 84 shows the factors in a drop-down form where prime factors are the last component. This image shows the factor tree of 84:

According to the factor tree,

84 = 4 × 21 = (2 × 2) × (3 × 7)

Thus, in the factor tree of 84 in the image: 2, 2, 3 and 7 come at the bottom most level while 84 is at the top.

Solved Examples

Example 1: Show the prime factorization of the number 84.

Solution: 

84 ÷ 2 = 42
42 ÷ 2 = 21
21 ÷ 3 = 7
7 ÷ 7 = 1

Therefore, 84 = 2 x 2 x 3 x 7 => 84 = 2² x 3 x 7

Example 2: List the factors of number 84.

Solution:

1 x 84 = 84
2 x 42 = 84
3 x 28 = 84
4 x 21 = 84
6 x 14 = 84
7 x 12 = 84

Therefore, the factors of number 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84.

Additionally, here are some fun facts about the factors of 84:

• 84 is a sum of twin prime numbers 41 and 43

• 84 is a tetrahedral number i.e. it makes a triangle

• 84 is the sum of the first 7 triangular numbers