How to Find Factors of 108 Step by Step
The concept of factors of 108 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding the factors of a number like 108 helps students solve divisor, LCM, HCF, and grouping problems quickly and accurately. On this page, you’ll learn what the factors of 108 are, how to find them, and tips and tricks for quick calculation.
What Is Factors of 108?
A factor of 108 is any whole number that divides 108 exactly with no remainder. In other words, if you multiply two whole numbers and get 108 as the answer, both of those numbers are factors of 108. You’ll find this concept applied in areas such as fraction simplification, LCM and HCF calculation, and solving divisibility problems.
Key Formula for Factors of 108
Here’s the standard factorization formula: \( 108 = 2^2 \times 3^3 \).
This means the prime factors of 108 are 2 and 3, and you can build all the other factors using these two numbers in various combinations.
How to Find Factors of 108
To find all factors of 108, start with 1 and go up to 108. For each number, check if 108 divided by that number gives a whole number answer. If it does, both numbers in that division form a factor pair of 108. This approach is called the division method.
- Start with 1:108 ÷ 1 = 108 (factors: 1, 108)
- Try 2:108 ÷ 2 = 54 (factors: 2, 54)
- Continue with 3:108 ÷ 3 = 36 (factors: 3, 36)
- Continue checking each number up to 108.
Factors of 108 – List and Pairs
By following the stepwise division method above, here is a complete table of the factors of 108 and their pairs. This helps you spot patterns and quickly recall answers during tests and problem-solving.
| Factor | Pair Factor | Explanation |
|---|---|---|
| 1 | 108 | 1 × 108 = 108 |
| 2 | 54 | 2 × 54 = 108 |
| 3 | 36 | 3 × 36 = 108 |
| 4 | 27 | 4 × 27 = 108 |
| 6 | 18 | 6 × 18 = 108 |
| 9 | 12 | 9 × 12 = 108 |
So, the full list of factors of 108 is:
1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108
Prime Factorization of 108
Prime factorization breaks 108 into a product of only prime numbers. Here is the step-by-step prime factorization of 108:
- Divide 108 by 2: 108 ÷ 2 = 54
- Divide 54 by 2: 54 ÷ 2 = 27
- Divide 27 by 3: 27 ÷ 3 = 9
- Divide 9 by 3: 9 ÷ 3 = 3
- Divide 3 by 3: 3 ÷ 3 = 1
So, 108 = 2 × 2 × 3 × 3 × 3 = \(2^2 \times 3^3\)
Negative and Positive Factors of 108
Every positive factor of 108 has a corresponding negative factor because multiplying two negative numbers gives a positive result. So, the negative factors of 108 are:
-1, -2, -3, -4, -6, -9, -12, -18, -27, -36, -54, -108
This means 108 has 12 positive and 12 negative factors in total.
Properties & Everyday Uses of Factors of 108
The factors of 108 are useful in:
- Splitting 108 objects into equal groups (e.g., teams, seating).
- Finding the greatest common factor (GCF) or least common multiple (LCM) with other numbers—for example, with factors of 54.
- Solving word problems that involve divisibility, grouping, and distributing items evenly.
- Practical questions in exams: “How many ways can you arrange 108 chairs in equal rows?”
- Prime factorization skills help us simplify large number calculations, a useful tool in competitive tests.
Speed Trick or Vedic Shortcut
Here’s a quick tip for prime factorization: Divide repeatedly by the smallest available prime until you reach 1. Write each division step, so you avoid missing out on any factor. This “division ladder” method is a proven time-saver for exams. Vedantu’s teachers often explain such methods in live classes to boost calculation speed and precision.
Try These Yourself
- List all the factors of 108 between 15 and 40.
- Which is the largest prime factor of 108?
- Write all factor pairs of 108 using both positive and negative numbers.
- Is 54 a factor or a multiple of 108?
Common Mistakes and Misunderstandings
- Forgetting negative factors when asked for all possible factors.
- Missing out factor pairs (e.g., 9 × 12).
- Mixing up factors and multiples—remember, factors divide the number exactly.
Relation to Other Concepts
Understanding factors of 108 helps build a strong base for LCM and HCF, factor trees, and properties of composite numbers. For example, prime factorization is vital for understanding how numbers break down into basic building blocks. Try working out factors of 72 and factors of 120 to reinforce these ideas.
Quick Facts Table: All About Factors of 108
| Property | Value |
|---|---|
| Total Positive Factors | 12 |
| Total Negative Factors | 12 |
| Prime Factors | 2, 3 |
| Prime Factorization | 2 × 2 × 3 × 3 × 3 |
| Sum of Factors | 280 |
| Product of Factors | (108)6 |
Practice Questions with Solutions
- How many positive even factors does 108 have?
Positive even factors: 2, 4, 6, 12, 18, 36, 54, 108.
Total = 8. - Find all factor pairs of 108 where both numbers are less than 30.
Pairs: (6, 18), (9, 12), (12, 9), (18, 6) - Is 27 a factor of 108? Show calculation.
108 ÷ 27 = 4 (No remainder). Yes, 27 is a factor of 108. - Write the prime factorization and sum of all factors of 108.
Prime factorization: 22 × 33.
Sum: 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 + 27 + 36 + 54 + 108 = 280.
Classroom Tip
A quick way to check if a number is a factor: Divide 108 by it and see if there’s no remainder. Factoring games and grouping exercises make this topic fun and memorable. Vedantu’s academic experts use visuals and practice quizzes to help you remember patterns.
We explored factors of 108—from definition, calculation, prime factors, to practical uses and smart shortcuts. Continue practicing with Vedantu’s advanced maths resources to boost confidence for school and exam success.
Related Concepts: Explore More
FAQs on Factors of 108: Complete Guide with Pairs & Prime Factorization
1. What are the factors of 108?
The factors of 108 are all the numbers that divide 108 without leaving a remainder. These include both positive and negative factors. The positive factors of 108 are: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108. The negative factors are simply the opposites of these numbers.
2. What is the prime factorization of 108?
The prime factorization of 108 is 22 x 33. This means that 108 can be expressed as the product of its prime factors: 2 x 2 x 3 x 3 x 3.
3. How many factors does 108 have?
108 has a total of 12 positive factors and 12 negative factors, making 24 factors in total. This is because each positive factor has a corresponding negative factor.
4. What are the factor pairs of 108?
The factor pairs of 108 are pairs of numbers that multiply to 108. These include: (1, 108), (2, 54), (3, 36), (4, 27), (6, 18), and (9, 12). Remember that each pair also has a negative counterpart (e.g., (-1, -108)).
5. How do I find the factors of 108?
To find the factors of 108, you can use the division method. Systematically divide 108 by integers starting from 1, and if the division results in a whole number (no remainder), that integer is a factor. Continue this process until you reach the square root of 108 (approximately 10.4), as factors will repeat after this point.
6. What is the sum of all the factors of 108?
The sum of all the factors of 108 (including both positive and negative factors) is 0. The sum of positive factors is 280, and the sum of negative factors is -280.
7. Is 108 a perfect square?
No, 108 is not a perfect square. A perfect square is a number that can be obtained by squaring an integer. The square root of 108 is not a whole number.
8. Is 108 a perfect cube?
No, 108 is not a perfect cube. A perfect cube is a number that can be obtained by cubing an integer. The cube root of 108 is not a whole number.
9. How can I use the factors of 108 to find the least common multiple (LCM) or greatest common factor (GCF)?
To find the LCM or GCF of 108 and another number, use the prime factorization of 108 (22 x 33). Compare these prime factors with the prime factors of the other number. The LCM includes all prime factors raised to their highest powers, while the GCF includes only common prime factors raised to their lowest powers.
10. What are some real-world applications of finding factors?
Finding factors is useful in various real-world situations, such as: dividing items into equal groups, determining dimensions of rectangles with a given area (e.g., arranging 108 tiles into a rectangular array), or solving problems involving ratios and proportions.
11. What is the difference between factors and multiples?
Factors are numbers that divide a given number without leaving a remainder, while multiples are numbers obtained by multiplying a given number by integers. For example, the factors of 108 are 1, 2, 3, etc., while the multiples of 108 are 108, 216, 324, etc.
12. Explain the concept of divisibility rules in relation to finding factors.
Divisibility rules provide shortcuts to determine if a number is divisible by another number without performing long division. For instance, if a number is even, it's divisible by 2; if the sum of its digits is divisible by 3, it's divisible by 3. Understanding divisibility rules can speed up the process of finding factors.
