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Introduction: Why Does 84 Deserve Your Attention?
Imagine you have 84 chocolates and want to share them equally among friends. Could you split them among 2 friends? 3? 7? 12? Each time the chocolates divide perfectly, you have just discovered a factor of 84.
The factors of 84 are all the numbers that divide 84 exactly, leaving no remainder. Whether you are a student tackling homework, a parent helping with maths, or simply a curious mind, this guide will walk you through everything about the factors of 84 in a clear and fun way.
What Are Factors, Anyway?
A factor of a number is a whole number that divides it completely. If 84 ÷ n gives a whole number with remainder 0, then n is a factor of 84. For example, 84 ÷ 7 = 12, so both 7 and 12 are factors.
Factors always come in teams: multiply two of them together, and you get back the original number.
All Factors of 84
Here are all factors of 84, listed from smallest to largest:
12 factors in total!
Quick Facts About 84
Here are some interesting things to know about this number:
- 84 has exactly 12 factors.
- The smallest factor is 1, and the largest is 84 itself.
- The sum of all the factors of 84 is 224.
- 84 is an even number, so 2 is always one of its factors.
How to Find the Factors of 84 (Step by Step)
Finding factors doesn't have to be complicated! Try the division method. Test each whole number starting from 1 and check whether it divides 84 evenly.
Let's Test the Numbers!
Divide 84 by each number and see which calculations give a whole number.
You can stop at 9, because the square root of 84 is about 9.17. Beyond that point, the factor partners you find simply repeat the ones you already have. This shortcut saves a lot of time.
Factors of 84 in Pairs
Pair factors are two numbers that multiply to give 84. Because 84 is not a perfect square, every factor has a distinct partner. The factors of 84 in pairs are:
That gives six factor pairs: (1, 84), (2, 42), (3, 28), (4, 21), (6, 14), and (7, 12).
Notice that the pairs get closer together as you move down the list. The closest pair, 7 and 12, sits right around the square root of 84.
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Prime Factorization of 84
Every number is built from prime numbers, and 84 is no exception. Let's break it down using a simple factor tree and discover its prime building blocks!
The Factor Tree of 84
Prime Factorization
Or, using exponents: 2² × 3 × 7
So the prime factorization of 84 is 2 × 2 × 3 × 7, or written with exponents, 2² × 3 × 7. The prime factors of 84 are 2, 3, and 7.
A Neat Trick to Count Factors Without Listing Them
Take the exponents in the prime factorization (2, 1, and 1), add 1 to each, and multiply:
This means 84 has exactly 12 factors. That matches our list of 12 factors exactly!
Factors of -84
Here's something many students overlook: factors can be negative too. Since a negative times a negative gives a positive, every positive factor of 84 has a negative twin. The factors of -84 are:
Strictly speaking, the factors of -84 include both the positive and negative numbers, so the complete set is ±1, ±2, ±3, ±4, ±6, ±7, ±12, ±14, ±21, ±28, ±42 and ±84, a total of 24 values.
The negative factor pairs of 84 are (-1, -84), (-2, -42), (-3, -28), (-4, -21), (-6, -14) and (-7, -12), since the product of two negatives is positive 84.
For 84 itself, pairs mix one positive and one negative number, for example 1 × (-84) or (-7) × 12.
Divisibility Rules That Prove 84 Is So Flexible
Let's see why 84 is divisible by several numbers using these simple divisibility rules:
Divisible by 2: It ends in an even digit (4).
Divisible by 3: 8 + 4 = 12, and 12 is divisible by 3.
Divisible by 4: The last two digits are 84, which is divisible by 4.
Divisible by 6: It is divisible by both 2 and 3.
Divisible by 7: 7 × 12 = 84.
Prime factorization helps you find all the factors of a number, count them quickly, and understand how different numbers divide evenly. For 84, the prime building blocks are 2, 3, and 7!
Quick Practice Questions
Time to test your understanding of the factors of 84! Try answering these questions.
How many factors does 84 have?
Answer: 12What is the greatest common factor of 84 and 36?
Answer: 12Which factor pair of 84 has the smallest difference?
Answer: 7 and 12Is 5 a factor of 84?
Answer: No, 84 ÷ 5 = 16.8Conclusion
The factors of 84, namely 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84, reveal how neatly this number can be divided and rearranged. With six factor pairs, a prime factorization of 2² × 3 × 7, and 12 factors in total, 84 is a wonderfully versatile number. Master the division method, remember the pair trick, and you will find factors of any number with confidence.
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