What are the Factor Pairs of 120?
The concept of factors of 120 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Learning how to find factors of 120 quickly builds number sense, helps in LCM/HCF questions, and improves calculation speed for competitive exams and school assessments.
What Are Factors of 120?
A factor of 120 is a whole number that divides 120 exactly, leaving no remainder. In other words, if you multiply two whole numbers and get 120, both those numbers are factors. This is useful for understanding composite numbers, finding common factors, and solving arithmetic or algebraic problems involving divisibility.
Complete List: The factors of 120 are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120.
Key Formula for Factors of 120
Here’s the standard formula using prime factors: \( 120 = 2^3 × 3^1 × 5^1 \)
All factors can be generated by taking all possible products of powers of 2 (0 to 3), 3 (0 to 1), and 5 (0 to 1).
Prime Factorization of 120
Prime factors of 120 are the building blocks for the number. Breaking 120 into only prime number multipliers lets us see its structure clearly.
- Divide 120 by 2: 120 ÷ 2 = 60
- 60 ÷ 2 = 30
- 30 ÷ 2 = 15
- 15 ÷ 3 = 5
- 5 ÷ 5 = 1
So, the prime factors of 120 are 2 × 2 × 2 × 3 × 5 or more compactly, 23 × 3 × 5.
Factor Pairs of 120
A factor pair of 120 consists of two whole numbers whose product is 120. For MCQs and mental maths, pair listing helps avoid missing factors.
| Factor 1 | Factor 2 | Check |
|---|---|---|
| 1 | 120 | 1 × 120 = 120 |
| 2 | 60 | 2 × 60 = 120 |
| 3 | 40 | 3 × 40 = 120 |
| 4 | 30 | 4 × 30 = 120 |
| 5 | 24 | 5 × 24 = 120 |
| 6 | 20 | 6 × 20 = 120 |
| 8 | 15 | 8 × 15 = 120 |
| 10 | 12 | 10 × 12 = 120 |
How to Find Factors of 120 (Stepwise Method)
- Start with 1 and 120 (since 1 × 120 = 120)
- Test every number from 2 up to the square root of 120 (about 10.95).
For each number n, if 120 ÷ n is whole, then n and 120÷n are both factors. - List these as factor pairs to avoid duplication and ensure you do not miss any.
For practice, follow the same method with another number, like the factors of 60.
Properties and Types of Factors of 120
Even factors: 2, 4, 6, 8, 10, 12, 20, 24, 30, 40, 60, 120
Odd factors: 1, 3, 5, 15
Prime factors: 2, 3, 5
Composite factors: 4, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
Universal factors: 1 and 120 (the smallest and largest possible factors of 120).
Remember, all factors (except 1 and 120) are composite, since 120 is a composite number. For more on prime factors and composite numbers, check out those pages for definitions and examples.
Prime Factorization Tree of 120
Using a factor tree, you can visualize the prime decomposition:
- 120 breaks into 2 × 60
- 60 breaks into 2 × 30
- 30 breaks into 2 × 15
- 15 breaks into 3 × 5 (both prime)
So the full breakdown is 2 × 2 × 2 × 3 × 5. This tree method is especially useful in factorization and LCM/HCF questions.
Speed Trick or Vedic Shortcut
Here's a fast way to find factors of any number like 120: List 1 and the number, then keep checking consecutive numbers (2, 3, 4...) up to the square root, pairing each with its complement. For timed exams, write pairs vertically to avoid repeats!
Example: Check if 7 divides 120: 120 ÷ 7 = 17.14 (not a whole number, so 7 is not a factor). If the division gives a decimal, skip to the next. This trick works for all numbers.
Try These Yourself
- Write all factor pairs of 120 including negative pairs.
- Check if 30 and 24 are factors of 120.
- Find the common factors of 60 and 120. (Tip: Use the common factors tool.)
- Write the sum of all factors of 120.
Solved Examples
Example 1: Which pair of factors of 120 add up to 23?
1. List factor pairs: (1,120), (2,60), (3,40), (4,30), (5,24), (6,20), (8,15), (10,12)
2. Check which sum is 23: 8 + 15 = 23
Final Answer: (8,15) is the pair.
Example 2: What is the highest common factor (HCF) of 90 and 120?
1. Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
2. Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
3. Common: 1, 2, 3, 5, 6, 10, 15, 30
4. HCF = 30
Example 3: What are the prime factors of 120?
120 = 2 × 2 × 2 × 3 × 5 = 23 × 3 × 5
Frequent Errors and Misunderstandings
- Missing factor pairs by stopping at 10 (must check all up to square root of 120).
- Confusing factors and multiples—remember, factors divide 120, while multiples are products like 240, 360, etc.
- Ignoring 1 and 120 (always include smallest and largest for completeness).
Relation to Other Concepts
The idea of factors of 120 connects with LCM and HCF, multiplication tables, and the table of 20. Mastering factors supports algebra and number theory in higher classes, and also helps in daily logical reasoning.
Classroom Tip
To quickly find all factors of 120, list pairs systematically: Start with 1, then try 2, 3, 4 ..., checking for no remainder. Vedantu’s teachers use visual tables and short tricks in live classes to encourage stepwise, error-free thinking.
We explored factors of 120—from basics, listing, prime factorization, tricks, to connections with LCM and practice examples. Continue learning and practicing with Vedantu to build confidence in math and ace competitive exams!
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FAQs on Factors of 120: Definition, Prime Factors & Pairs
1. What are the factors of 120?
The factors of 120 are all the whole numbers that divide 120 exactly without leaving a remainder. These are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120. Remember that factors can also be negative, so -1, -2, -3... -120 are also factors.
2. How many factors does 120 have?
120 has a total of 16 positive factors and 16 negative factors, making a total of 32 factors.
3. What is the prime factorization of 120?
The prime factorization of 120 is 23 × 3 × 5. This means that 120 can be expressed as the product of its prime factors: 2 multiplied by itself three times, then multiplied by 3, and finally multiplied by 5.
4. What are the factor pairs of 120?
Factor pairs of 120 are pairs of numbers that, when multiplied together, equal 120. These pairs are: (1, 120), (2, 60), (3, 40), (4, 30), (5, 24), (6, 20), (8, 15), and (10, 12). Negative factor pairs also exist, such as (-1, -120), (-2, -60), etc.
5. How do I find the factors of 120 using the division method?
To find the factors using division, systematically divide 120 by each whole number, starting from 1. If the division results in a whole number (no remainder), both the divisor and the quotient are factors. Continue this process until the quotient becomes less than the divisor.
6. What is the difference between factors and multiples?
Factors are numbers that divide a given number evenly (with no remainder), while multiples are numbers obtained by multiplying a given number by whole numbers. For example, the factors of 120 include 1, 2, 3, etc., while the multiples of 120 are 120, 240, 360, etc.
7. Is 120 a composite number?
Yes, 120 is a composite number because it has more than two factors (1 and itself).
8. How are factors of 120 useful in finding the LCM and HCF?
Finding the prime factorization of numbers (like 120) is crucial for determining the Least Common Multiple (LCM) and Highest Common Factor (HCF) of two or more numbers. The LCM is the smallest number that is a multiple of all the given numbers, while the HCF is the largest number that is a factor of all the given numbers.
9. What are some common mistakes students make when finding factors?
Common mistakes include: forgetting 1 and the number itself as factors; missing factors during systematic division; confusing factors with multiples; and not considering negative factors.
10. Are there any shortcuts or tricks for finding factors quickly?
One trick is to find the prime factorization first. Then, systematically combine these prime factors to generate all possible factors. Using divisibility rules can also speed up the process.
11. What are the even and odd factors of 120?
The even factors of 120 are: 2, 4, 6, 8, 10, 12, 20, 24, 30, 40, 60, and 120. The odd factors are: 1, 3, 5, and 15.
12. How can I use a factor tree to find the prime factorization of 120?
Start with 120. Branch it into two factors (e.g., 2 and 60). Continue branching until all branches end in prime numbers. The prime factorization will be the product of all the prime numbers at the end of the branches. (For 120, this would result in 2 x 2 x 2 x 3 x 5).
