How to Find Factors 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 multiplying. 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
What are the Practical Applications of the Factorization Equations in Our Day-to-Day Lives?
Being efficient at solving equations involving factorization principles can be helpful for the common people in carrying out different tasks in their daily lives. Mastering the skill of factoring a particular numeric value can help people to get a quick understanding of how to divide something or any numeric value into equal values. It can help them to calculate easily while buying something.
After buying something, people can quickly analyse how much money they should exchange with the shopkeeper if they are good at factorisation. The concept of factorization can help them in comparing the prices of different objects available on different platforms. Also, if they are good at factoring it will help them to have a detailed and quick analysis of the remaining time that they will need to reach a place.
All types of mathematical equations help people to build the habit of quick analysation and calculation. Every mathematical equation has some kind of practical usages and we need to understand the basic concepts of mathematics so that we do not make mistakes while exchanging money with other people.
FAQs on Factors of 84
1. How do I know what are the factors of 84?
There are three ways to see what are the factors of 84, i.e., by multiplying the numbers less than 84, prime factorization method, and to make a factor tree of 84.
2. How do I prime factorize 84?
Prime factorization of the number 84 is a simple process. Successively divide 84 by the smallest prime number to get a whole number. First divide 84 ÷ 2 = 42. Now 42 ÷ 2 = 21. Continuing this, 21 ÷ 2 ≠ a whole number, so you take the next prime number, i.e., 3; 21 ÷ 3 = 7. Then 7 ÷ 7 = 1, and your prime factorization of number 84 ends here. Eventually, you have 84 = 2 x 2 x 3 x 7 or 84 = 2² x 3 x 7.
3. How to quickly solve mathematical equations involving the factorization of a number?
Factorization of numbers is one of the basic concepts of mathematical equations and the students who want to score well in mathematics must understand the concept of factorization. In order to do factorization of a particular number, the students must be well familiarised with the formula that they should apply in quickly solving the problems. The concept of factorization is also included in algebra and to solve algebraic equations that involve factorization of a particular number the students must remember the most commonly used algebraic formulae.
Since applying the formula to solve algebraic equations might be difficult, the students must start to deal with the factorization problems by considering real numbers. Mathematical equations involving real numbers other than the variables are easier to calculate and once the students become quite efficient at solving the factorization equations then they can start practising with the algebraic variables. Also, they should start practising the techniques to simplify a mathematical equation so that applying the factorization formulae becomes easier for them. Simplifying the mathematical equations will also help the students to solve them quickly without consuming much time.
Remembering the formula of cross multiplication can also be helpful for the students in solving factoring equations. Applying the technique of cross multiplication will be particularly helpful for them if they are dealing with whole numbers. Other than that, the more they will practice solving the equations the more it will be easier for them to deal with the factorization problems.
4. Why is learning the concept of factorization important?
Factorization problems help us to find out the value of numeric that is otherwise difficult to figure out. There are various types of factorisation and one of them is the greatest common factor. The students can also apply the formula of the grouping method, square method. They can also use the technique of figuring out the sum or the difference in cubes in order to deal with factorization equations.
The factorization equations help us to write the polynomials in a simple and easy to calculate format. When we simplify mathematical equations and apply the correct principles, it becomes easier for us to solve the equations and to find out the exact value that we are looking for. Factorization is important because it helps the students to have a good grip over the various methodologies and techniques to solve mathematical equations.
5. How to understand the concept of factorization easily?
Learning the concept of factorization is not only important for the students to score more in the examinations but also will help them to calculate in an easier manner in their daily lives.
The concept of factorization is not only applied to solve mathematical equations because we face a lot of situations in our day-to-day lives where we have to apply the law of factorization in order to calculate. To understand the concept easily the students should start with whole numbers and practice applying the formula of factorization. Gradually they should start solving algebraic equations where the factorization formula can be applied.
Also, the students can take help of the articles published on the website of Vedantu. Individual chapters that are included in their mathematics syllabus are thoroughly discussed and all the important facts and information are clearly mentioned in the study materials that the students need to remember. They can download the study materials for free and get access to several other reference books and revision notes to better understand the concepts. The details of the registration process are also available on the website of Vedantu so that the students can join the free classes and talk to the teachers directly.