If a number completely divides 84 without leaving a remainder, then that number is called a factor of 84. Thus, we can say that the factors of 84 are the divisors of 84. Here, we will learn about the factors of 84 and the prime factors of 84. We have:
- 84 = 2×2×3×7 is the prime factorization of 84
- The factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84
- The prime factors of 84 are 2, 3 and 7
- Negative factors of 84 are –1, -2, -3, -4, -6, -7, -12, -14, -21, -28, -42 and -84
Table of Contents
What are the Factors of 84?
Let us write the number 84 multiplicatively in all possible ways. We definitely have:
| 84 = 1×84 | ∴ 1, 84 are factors of 84 |
| 84 = 2×42 | ∴ 2, 42 are factors of 84 |
| 84 = 3×28 | ∴ 3, 28 are factors of 84 |
| 84 = 4×21 | ∴ 4, 21 are factors of 84 |
| 84 = 6×14 | ∴ 6, 14 are factors of 84 |
| 84 = 7×12 | ∴ 7, 12 are factors of 84 |
As there are no other ways we can write 84 multiplicatively, so we will stop right now. Note that the numbers in brown color are the factors of 84.
∴ we conclude that the factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.
It is known that if m is a factor of 84, then -m is also a factor of 84. So all the negative factors of 84 are -1, -2, -3, -4, -6, -7, -12, -14, -21, -28, -42, and -84.
Pair Factors of 84
| 84 = a×b | Pair Factors of 84 |
| 84 = 1×84 | (1, 84) |
| 84 = 2×44 | (2, 44) |
| 84 = 3×28 | (3, 28) |
| 84 = 4×21 | (4, 21) |
| 84 = 6×14 | (6, 14) |
| 84 = 7×12 | (7, 12) |
Thus the pair factors of 84 are (1, 84), (2, 42), (3, 28), (4, 21), (6, 14), and (7, 12).
Negative pair factors of 84: If (a, b) is a positive pair factor of 84, then (-a, -b) is a negative pair factor of 84. Thus, all the negative pair factors of 84 are (-1, -84), (-2, -42), (-3, -28), (-4, -21), (-6, -14), and (-7, -12).
Number of Factors of 84
From above we have calculated the factors of 84 which are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.
Thus the total number of factors of 84 is twelve.
Prime Factors of 84
Note that the factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84. Among those factors, we observe that only 2, 3, and 7 are prime numbers as they do not have any proper divisors.
∴ the prime factors of 84 are 2, 3, and 7.
Question: What are the factors of 84?
Video Solution:
Also Read:
| Factors of 50: | 1, 2, 5, 10, 25, 50 |
| Factors of 64: | 1, 2, 4, 8, 16, 32, 64 |
| Factors of 70: | 1, 2, 5, 7, 10, 14, 35, 70 |
| Factors of 72: | 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 |
How to Find Factors of 84?
Now we will determine the factors of 84 by division method. In this method, we will try to find the numbers that can divide 84 with no remainder. See that
| 84/1=84 and the remainder is 0 | ∴ 1 and 84 are factors of 84 |
| 84/2=42 and the remainder is 0 | ∴ 2 and 42 are factors of 84 |
| 84/3=28 and the remainder is 0 | ∴ 3 and 28 are factors of 84 |
| 84/4=21 and the remainder is 0 | ∴ 4 and 21 are factors of 84 |
| 84/6=14 and the remainder is 0 | ∴ 6 and 14 are factors of 84 |
| 84/7=12 and the remainder is 0 | ∴ 7 and 12 are factors of 84 |
Note that no numbers other than the numbers in violet color can divide 84. So the numbers in violet color, that is, 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84. are the complete list of factors of 84.
This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.