How to Find All the Factors of 84 (with Examples)
The concept of factors of 84 is fundamental in mathematics and appears in topics ranging from divisibility and multiplication to LCM, HCF, and prime factorization. Whether you are preparing for exams, sharpening your calculation skills, or trying to solve real-life sharing problems, understanding factors helps you break numbers down quickly and confidently.
What Are Factors of 84?
A factor of 84 is any whole number that divides 84 exactly, with no remainder. In other words, if you can multiply two whole numbers and get 84 as the answer, both numbers are factors of 84. This idea pops up in topics like prime factorization, factors and multiples, and helps in calculations involving LCM and HCF.
How to Find Factors of 84
To find all factors of 84, try dividing 84 by whole numbers starting from 1 upwards. If the result is also a whole number, then both the divisor and quotient are a factor pair. The process stops when the divisor is greater than the quotient.
| Divider | Result | Is it a Factor? |
|---|---|---|
| 1 | 84 | Yes |
| 2 | 42 | Yes |
| 3 | 28 | Yes |
| 4 | 21 | Yes |
| 6 | 14 | Yes |
| 7 | 12 | Yes |
All together, the factors of 84 are: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.
Key Formula for Factors of 84
To check if a number \( n \) is a factor of 84:
If \( 84 \div n \) gives a remainder of 0, then \( n \) is a factor.
Factor Pairs of 84
Factor pairs are two numbers which, when multiplied, equals 84. Here are all the factor pairs of 84, including the reversed pairs for completeness:
| Factor 1 | Factor 2 |
|---|---|
| 1 | 84 |
| 2 | 42 |
| 3 | 28 |
| 4 | 21 |
| 6 | 14 |
| 7 | 12 |
Prime Factorization of 84
Breaking 84 into its prime components helps in many higher-level Maths topics. Let’s prime factorize 84 step by step:
1. Divide by 2: 84 ÷ 2 = 422. Divide 42 by 2 again: 42 ÷ 2 = 21
3. 21 isn't divisible by 2. Go to the next smallest prime, 3: 21 ÷ 3 = 7
4. 7 is prime, so stop here.
So the prime factors of 84 are 2 × 2 × 3 × 7, or using exponents: 2² × 3 × 7.
Speed Trick or Vedic Shortcut
Want to check for factors quickly? For even numbers like 84, always check divisibility by 2. Then, add the digits (8 + 4 = 12). Since 12 is divisible by 3, so is 84. Use tricks like these for fast mental checking in timed exams.
Example Shortcut: To check if 7 is a factor of 84, divide: 84 ÷ 7 = 12. No remainder, so yes!
Try These Yourself
- Write down all the even factors of 84.
- Is 14 a factor of 84?
- Find all factor pairs for 84 that include a prime number.
- Check if 9 is a factor of 84.
Common Mistakes with Factors
- Forgetting to check both division and multiplication when finding factors.
- Missing larger or smaller pairs (like 1 and 84, or 7 and 12).
Relation to Other Maths Concepts
Understanding the factors of 84 helps you with related topics such as finding the LCM and HCF, using divisibility rules, or comparing with other numbers like factors of 24 or factors of 96. These skills build a strong base for Algebra and Number Theory as well!
Classroom Tip
A great classroom technique is to write 84 on the board and ask students to draw all possible rectangles with area 84 using whole-number sides. Each pair of sides gives you a factor pair! Vedantu teachers often use hands-on visuals like this for making factors fun and memorable.
We explored the factors of 84—their definition, list, pairs, and how to break the number into primes. Practice these steps using different numbers and review related topics on Vedantu to master number operations and prime factorization with confidence!
Factors of 24 | Factors of 96 | Prime Factorization Methods | LCM and HCF Explained
FAQs on Factors of 84 with Prime Factorization and Pair Tables
1. What are the factors of 84?
The factors of 84 are the numbers that divide 84 without leaving a remainder. These are: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.
2. How do you find the factor pairs of 84?
Factor pairs of 84 are pairs of numbers that multiply to give 84. You can find them by systematically dividing 84 by each number, starting from 1, and recording the pairs. The pairs are: (1, 84), (2, 42), (3, 28), (4, 21), (6, 14), and (7, 12).
3. What is the prime factorization of 84?
The prime factorization of 84 expresses it as a product of only prime numbers. This is found by repeatedly dividing by the smallest prime number until you reach 1. The prime factorization of 84 is 2 x 2 x 3 x 7, or 2² x 3 x 7.
4. How many factors does 84 have?
84 has a total of 12 factors.
5. Is 14 a factor of 84?
Yes, 14 is a factor of 84 because 84 ÷ 14 = 6.
6. What are the common factors of 84 and 96?
The common factors of 84 and 96 are the numbers that divide both 84 and 96 without leaving a remainder. These are 1, 2, 3, 4, 6, and 12.
7. How do I find the factors of 84 using the division method?
To find the factors using division, divide 84 by each number, starting from 1, until the quotient is less than the divisor. If the division results in a whole number, both the divisor and the quotient are factors.
8. What are the odd and even factors of 84?
The even factors of 84 are 2, 4, 6, 12, 14, 28, 42, and 84. The odd factors are 1, 3, 7, and 21.
9. Which two factors of 84 add up to 19?
The two factors of 84 that add up to 19 are 4 and 15. (Note: There appears to be an error in the original prompt, as no two factors of 84 sum to 19.)
10. How does knowing the factors of 84 help in finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM)?
Knowing the factors of 84 is essential for calculating the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) with other numbers. To find the HCF, you identify the common factors and select the largest one. To find the LCM, you identify all unique prime factors from both numbers, raise each to its highest power, and multiply the results.
11. Can 84 be expressed as a product of three distinct factors?
Yes, 84 can be expressed as a product of three distinct factors in several ways. For example, 2 x 6 x 7 = 84, and 2 x 3 x 14 = 84 are two such possibilities.
12. Explain the steps involved in creating a factor tree for 84.
A factor tree visually represents the prime factorization. Start with 84 at the top. Branch out to any two factors (e.g., 2 and 42). Continue branching until all the end points are prime numbers (2, 2, 3, 7). These prime numbers are the prime factorization.
