All factors of 84 include 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84. The factors of 84 are the numbers that can divide 84 with no remainder left. These factors include both positive and negative integers, as well as 1 and 84.

In this article, we will discuss all the topics related to the factors of 84, including all the factors of 84, both prime factors, and methods to find these factors.

What are Factors?

The factor of a number is any number that divides the given number completely without leaving any remainder.

In the case of 84, all the numbers like 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84 are the factors of 84 because they all divide 84 completely without leaving any remainder.

Learn, Factors

What are Factors of 84?

The factors of 84 are those numbers that can evenly divide 84 without leaving a remainder. For example, 84 can be divided by 2 with quotient of 42. Similarly.

• 3 is the Factor of 84 as 84/3 = 24.
• 4 is a factor of 84 as 84 ÷ 4 = 21.
• 6 is a factor of 84 as 84 ÷ 6 = 14.
• 7 is a factor of 84 as 84 ÷ 7 = 12.
• 12 is a factor of 84 as 84 ÷ 12 = 7.
• 14 is a factor of 84 as 84 ÷ 14 = 6.
• 21 is a factor of 84 as 84 ÷ 21 = 4.
• 28 is a factor of 84 as 84 ÷ 28 = 3.
• 42 is a factor of 84 as 84 ÷ 42 = 2.
• 84 is a factor of 84 as 84 ÷ 84 = 1.

All Factors of 84

The factors of 84 are: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.

Prime Factors of 84

List of all factor of 84 include both prime and composite numbers. However, we can also express 84 as the product of only prime numbers. The numbers 2 x 2 x 3 x 7 that multiply together to give back the number 84, are called the prime factors of 84.

How to Find the Factors of 84?

To find the factors of 84, you can use the hit and try method to check if a certain number completely divides 84 or not, but before that keep the following things in mind:

• The number 1 and the number itself are always the factors of the given number.
• The factors of a number are always less than the given number.

Now, that we know that 1 and 84 are the factors of 84, we can start checking numbers starting from 2.

Check for 2: 2 completely divides 84, hence, 2 is a factor of 84.

• 2 × 42 = 84

Check for 3: 3 completely divides 84, hence, 3 is a factor of 84.

• 3 × 28 = 84

Check for 4: 4 completely divides 84, hence, 4 is a factor of 84.

• 4 × 21 = 84

Check for 5: 5 doesn’t completely divide 84, so 5 is not a factor of 84.

Check for 6: 6 completely divides 84, hence, 6 is a factor of 84.

• 6 × 14 = 84

Check for 7: 7 completely divides 84, so 7 is a factor of 84.

• 7 × 12 = 84

Check for 8: 8 doesn’t completely divide 84, so 8 is not a factor of 84.

Check for 9: 9 doesn’t completely divide 84, so 9 is not a factor of 84.

You can keep checking further in the same way but here is a little trick to save you time because checking every number until 84 is not feasible:

By far, these are the factor pairs that we have found out:

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

The next number after 7 that completely divides 84 is 12 with the following factor pair:

12 × 7 = 84

You can see that this factor pair is the exact reverse of the the previous factor pair:

7 × 12 = 84

This is exactly the point where you can stop checking for further factors because if we take all the values of these factor pairs, we will get all the factors of 84.

In this case, the factors are: 1, 2, 3, 4, 6, 7, 12, 14, 28, 42, and 84.

Prime Factorization of 84

To express 84 as the factor of prime numbers, use prime factorization. Prime factorization is a process of repeated division such that the divisor each time is only a prime number. For 84,

• 84 / 2 = 42
• 42 / 2 = 21
• 21 / 3 = 7
• 7 / 7 = 1

You can see here that in each step, we have only used a prime number for division until we completely divide 84. This can be represented as follows:

Read More about Prime Factorization.

Factor Tree of 84

Factor tree is a chart that shows all the prime factors of a number in the form of a tree like diagram, for 84 we can make the factor tree of 84 as follows:

Factor Pairs of 84

The factors when paired together, give the original number back. These pairs can include both positive and negative numbers.

Positive Factor Pairs of 84

The positive factor pairs of 84 are:

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

Negative Factor Pairs of 84

The negative factor pairs of 84 are:

• (-1) × (-84) = 84
• (-2) × (-42) = 84
• (-3) × (-28) = 84
• (-4) × (-21) = 84
• (-6) × (-14) = 84
• (-7) × (-12) = 84

Read More,

• Factors of 20
• Factors of 60
• Factors of 28
• Factors of 36
• Factors of 25
• Factors of 21

Solved Examples on Factors of 84

Example 1: Find the common factors of 84 and 114.

Solution:

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

Factors of 114 = 1, 2, 3, 6, 19, 38, 57, 114

Common factors = 1, 2, 3 and 6

Example 2: Is 84 an abundant number?

Solution:

The factors of 84 are = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

The sum of these factors, excluding 84 is = 140

Thus, since 84 is smaller than the sum of its factors, excluding 84 itself, yes, it is an abundant number.

Example 3: Are factors and multiples of 84 same?

Solution:

No, the factors of 84 are the numbers smaller than or equal to 84, that completely divide 84 whereas, the multiples are the numbers that come in the table of 84 and are equal to and greater than 84.

Practice Problems on Factors of 84

Problem 1: What is the greatest prime factor of 84?

Problem 2: What will be the value of a, if a x b = 84 and b = 42?

Problem 3: For what highest value of a, will a x b = 84?

Problem 4: Find the common factors of 84 and 116.

Problem 5: Are there any common numbers in the factors and multiples of 84?

Factors of 84 Frequently Asked Questions

List all Factors of 84.

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

How do you Find the Factors of 84?

To find the factors of 84, you can divide 84 by various numbers and check which ones result in whole number quotients without remainders.

What is the Prime Factorization of 84?

The prime factorization of 84 is 2² × 3 × 7.

What is the greatest common factor (GCF) of 84 and 36?

The GCF of 84 and 36 is 12.

Can 84 be evenly divided by 5?

No, 84 cannot be evenly divided by 5.

What is the Sum of All the Factors of 84?

The sum of all the factors of 84 is 224.

Is 84 a Perfect Square?

No, 84 is not a perfect square because it cannot be expressed as the square of an integer.