What Are the Factors of 60 List Prime Factorization and Factor Pairs
The concept of factors of 60 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Students often encounter this concept in topics such as divisibility, prime factorization, HCF & LCM, and problem-solving for competitive exams. Let’s explore everything you need to master the factors of 60 with simple explanations, clear steps, solved examples, and handy tips for fast calculation!
What Is Factors of 60?
A factor of 60 is any whole number that divides 60 exactly, without leaving a remainder. In other words, multiplying any of these numbers with another whole number gives you the product 60. This concept is often linked to prime factorization, factors of a number, and multiples of 60.
How to Find Factors of 60
To find all the factors of 60, simply list all pairs of whole numbers whose product is 60. Only numbers that divide 60 without leaving any remainder are included.
-
Divide 60 by all numbers from 1 up to 60.
If there’s no remainder, that number is a factor. -
List each result as a factor and form pairs.
Continue until pairs repeat.
| Division | Quotient | Is a Factor? |
|---|---|---|
| 60 ÷ 1 | 60 | Yes |
| 60 ÷ 2 | 30 | Yes |
| 60 ÷ 3 | 20 | Yes |
| 60 ÷ 4 | 15 | Yes |
| 60 ÷ 5 | 12 | Yes |
| 60 ÷ 6 | 10 | Yes |
| 60 ÷ 10 | 6 | Yes |
| 60 ÷ 12 | 5 | Yes |
| 60 ÷ 15 | 4 | Yes |
| 60 ÷ 20 | 3 | Yes |
| 60 ÷ 30 | 2 | Yes |
| 60 ÷ 60 | 1 | Yes |
Complete List of Factors & Factor Pairs of 60
All positive factors of 60 are:
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
The factor pairs of 60 (numbers that multiply to give 60) can be shown in a simple table:
| Factor 1 | Factor 2 | Pair Product |
|---|---|---|
| 1 | 60 | 1 × 60 = 60 |
| 2 | 30 | 2 × 30 = 60 |
| 3 | 20 | 3 × 20 = 60 |
| 4 | 15 | 4 × 15 = 60 |
| 5 | 12 | 5 × 12 = 60 |
| 6 | 10 | 6 × 10 = 60 |
Prime Factorization of 60
Prime factorization means expressing 60 as a product of prime numbers only. Let’s break it down step by step:
1. 60 ÷ 2 = 302. 30 ÷ 2 = 15
3. 15 ÷ 3 = 5
4. 5 is prime, so the process stops.
So, the prime factorization of 60 = 2 × 2 × 3 × 5 or 22 × 3 × 5.
A factor tree visually represents this breakdown. For a deep dive, see this prime factor tree example.
Properties & Applications of Factors of 60
- 60 is a composite number (it has more than two factors).
- It is an even number; so 2 is always a factor.
- Applications include finding the HCF, LCM, simplifying ratios, and solving real-world math problems.
You’ll see the factors of 60 often appear in questions about divisibility rules, arranging items in equal groups, or finding compatible numbers in the divisibility rules topic.
Solved Examples: Factors of 60
Example 1: List all the positive factors of 60.
Answer: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
Example 2: What is the sum of the prime factors of 60?
Answer: Prime factors are 2, 3, and 5. Their sum is 2 + 3 + 5 = 10.
Example 3: Which pairs of factors multiplied equal 60, and both are less than 30?
Answer: (6, 10), (5, 12), (4, 15), (3, 20), (2, 30). (Note: only first four pairs have both numbers less than 30.)
Practice Worksheet: Factors of 60
| Question | Your Answer |
|---|---|
| Is 12 a factor of 60? | |
| List all the prime factors of 60. | |
| What is the smallest factor of 60 greater than 1? | |
| Find a factor of 60 that is also a factor of 24. | |
| Write the factor pair whose sum is 16. |
Check your answers and try more practice in Vedantu’s worksheets and classes!
Relation to Other Maths Concepts
Knowing the factors of 60 helps to build a strong foundation for topics like HCF and LCM, factors of 24, and factors of 36. This knowledge is also crucial for understanding multiples, prime factorization, and solving number puzzles efficiently.
Classroom Tip and Speed Shortcut
A simple trick to check if a number is a factor of 60 is to divide 60 by that number and see if the quotient is an integer. Quickly list out pairs by starting from 1 upwards, and stop once your first number repeats!
For more such calculation tricks, Vedantu teachers show Vedic maths shortcuts in live classes and worksheets.
We explored factors of 60—definition, listing, pairs, prime factorization, solved examples, and proven tricks. To build confidence, keep practicing and explore more topics through Vedantu. For related maths topics, you may learn about Factors of 48, Factors of 72, and Prime Factor Trees for deeper understanding.
FAQs on Factors of 60 Complete Guide with Definition and Examples
1. What are the factors of 60?
The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. A factor is a number that divides 60 exactly without leaving a remainder. For example, 60 ÷ 5 = 12 and 60 ÷ 12 = 5, so both 5 and 12 are factors of 60. Since 60 is a composite number, it has more than two factors.
2. How do you find the factors of 60?
You can find the factors of 60 by checking which numbers divide 60 exactly. Follow these steps:
- Start from 1 and test divisibility up to 60.
- List numbers that divide 60 without remainder.
- Pair each factor with its corresponding quotient.
3. What is the prime factorization of 60?
The prime factorization of 60 is 2² × 3 × 5. This means 60 can be expressed as the product of its prime factors:
- 60 ÷ 2 = 30
- 30 ÷ 2 = 15
- 15 ÷ 3 = 5
- 5 ÷ 5 = 1
4. How many factors does 60 have?
The number 60 has 12 factors. Using its prime factorization 2² × 3¹ × 5¹, apply the formula for total factors: (2+1)(1+1)(1+1) = 3 × 2 × 2 = 12. These include both 1 and 60.
5. What are the factor pairs of 60?
The factor pairs of 60 are pairs of numbers that multiply to give 60. They are:
- 1 × 60
- 2 × 30
- 3 × 20
- 4 × 15
- 5 × 12
- 6 × 10
6. Is 60 a composite number?
Yes, 60 is a composite number because it has more than two factors. A prime number has exactly two factors (1 and itself), but 60 has 12 factors. Therefore, it is not prime.
7. What are the common factors of 60 and 48?
The common factors of 60 and 48 are 1, 2, 3, 4, 6, and 12. These numbers divide both 60 and 48 exactly. The greatest common factor (GCF) among them is 12.
8. What is the greatest common factor (GCF) of 60 and 36?
The greatest common factor of 60 and 36 is 12. Using prime factorization:
- 60 = 2² × 3 × 5
- 36 = 2² × 3²
9. What are the multiples of 60?
The multiples of 60 are numbers obtained by multiplying 60 by whole numbers. Examples include:
- 60 × 1 = 60
- 60 × 2 = 120
- 60 × 3 = 180
- 60 × 4 = 240
10. What is the difference between factors and multiples of 60?
The factors of 60 are numbers that divide 60 exactly, while the multiples of 60 are numbers obtained by multiplying 60 by whole numbers. For example:
- Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
- Multiples of 60: 60, 120, 180, 240, ...
