Factors of 84

Factors of 84 are integers that can be divided evenly into 84. It has total 12 factors of which 84 is the biggest factor and the prime factors of 84 are 2, 3 and 7. The Prime Factorization of 84 is 2² × 3 × 7.

• Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84
• Negative Factors of 84: -1, -2, -3, -4, -6, -7, -12, -14, -21, -28, -42 and -84
• Prime Factors of 84: 2, 3, 7
• Prime Factorization of 84: 2 × 2 × 3 × 7 = 2² × 3 × 7
• Sum of Factors of 84: 224

What are the Factors of 84?

Factors of a given number are the numbers which divide the given number exactly without any remainder.
Hence, the factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84.

84 is strictly smaller than the sum of its factors (except 84).
1 + 2 + 3 + 4 + 6 + 7 + 12 + 14 + 21 + 28 + 42 = 140
So, 84 is known as an abundant number.

How to Calculate the Factors of 84?

Various methods such as prime factorization and the division method can be used to calculate the factors of 84. In prime factorization, we express 84 as a product of its prime factors and in the division method, we see what numbers divide 84 exactly without a remainder.

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

Explore factors using illustrations and interactive examples.

• Factors of 120 - The factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
• Factors of 24 - The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24
• Factors of 96 - The factors of 96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
• Factors of 72 - The factors of 72 are  1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
• Factors of 42 - The factors of 42 are 1, 2, 3, 6, 7, 14, 21, 42
• Factors of 60 - The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Factors of 84 by Prime Factorization

Prime factorization means expressing a number in terms of the product of its prime factors. We can do use the division method or factor tree to do this.

1. Prime Factorization by Division method

The number 84 is divided by the smallest prime number which divides 84 exactly, i.e. it leaves a remainder 0. The quotient is then divided by the smallest or second smallest prime number and the process continues till the quotient becomes indivisible.

Since 84 is even, it will be divisible by 2.
Let us divide 84 by the prime number 2

84 ÷ 2 = 42

42 is again even, let's again divide by 2

42 ÷ 2 = 21

Now, 21 is an odd number. So it won't be divisible by 2. Let's check the next prime number i.e. 3

21 ÷ 3 =7

7 is a prime number. So, it's not further divisible.

Prime factorization of 84 = 2 × 2 × 3 × 7

2. Prime Factorization by Factor Tree

The other way of prime factorization as taking 84 as the root, we create branches by dividing it by prime numbers. This method is similar to above division method. The difference lies in presenting the factorization.
The figure below shows the factor tree of 84. The composite numbers will have branches as they are further divisible. We continue making branches till we are left with only prime numbers.

The numbers inside the circles are the prime factors of 84

Now that we have done the prime factorization of our number, we can multiply them and get the other factors. Can you try and find out if all the factors are covered or not? And as you might have already guessed it, for prime numbers, there are no other factors.

Factors of 84 in Pairs

The factor pairs of a number are the two numbers which, when multiplied, give the required number.
Considering the number 84, we have

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

So, the factor pairs of 84 are : (1, 84), (2, 42), (3, 28), (4, 21), (6, 14), (7, 12)

The product of two negative numbers also results in a positive number i.e. (-ve) × (-ve) = (+ve).
So, (-1, -84), (-2, -42), (-3, -28), (-4, -21), (-6, -14), (-7, -12). are also factor pairs of 84.
Our focus in this article will be on the positive factors.

Important Notes:

• Every number has a finite number of factors
• The factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84
• As 84 is strictly smaller than the sum of its factors (leaving 84), it is called an abundant number.