How to Find All Factors and Factor Pairs of 60 Easily
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: Definition, Pairs, and Prime Factorization
1. What are the factors of 60?
The factors of 60 are the numbers that divide 60 exactly, leaving no remainder. These are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. Factors are whole numbers that can be multiplied together to produce another number.
2. How many factors does 60 have?
The number 60 has a total of 12 positive factors. This includes both the prime and composite factors.
3. What is the prime factorization of 60?
The prime factorization of 60 is 2 × 2 × 3 × 5, or 22 × 3 × 5. Prime factorization expresses a number as a product of its prime factors (numbers only divisible by 1 and themselves).
4. What are the factor pairs of 60?
The factor pairs of 60 are pairs of numbers that multiply to 60. These include: (1, 60), (2, 30), (3, 20), (4, 15), (5, 12), and (6, 10). Note that these are only the *positive* pairs; negative pairs also exist.
5. Is 60 a prime or composite number?
60 is a composite number because it has more than two factors. A prime number has only two factors: 1 and itself.
6. How do I find all the factors of 60?
To find all factors, systematically divide 60 by each whole number, starting from 1, noting those that divide evenly. Alternatively, list factor pairs. Begin with (1,60), then (2,30), and continue until the pairs repeat (or you reach the middle).
7. What are the prime factors of 60?
The prime factors of 60 are 2, 3, and 5. Remember that a prime factor is a prime number that is a factor of the given number.
8. What is a factor tree and how do I use it for 60?
A factor tree is a visual way to find the prime factorization of a number. For 60, you would start by splitting 60 into two factors (e.g., 6 and 10). Continue breaking down composite numbers into factors until you are left only with prime numbers. These are the prime factors.
9. What are some real-world applications of finding factors of 60?
Understanding factors is useful in various real-world situations, such as dividing items evenly among a group of people, calculating the dimensions of objects, and solving problems involving ratios and proportions. It’s a foundation for more advanced math.
10. What are the common factors of 60 and another number (e.g., 72)?
To find the common factors of 60 and 72, list the factors of each number and identify the numbers that appear in both lists. For example, the common factors of 60 and 72 are 1, 2, 3, 4, 6, and 12.
11. Can 60 be expressed as a sum of two of its factors?
Yes. For example, 60 can be expressed as the sum of 30 + 30 (both are factors of 60).
12. How can I quickly check if a number is a factor of 60?
The quickest way is to use division. If the division of 60 by that number results in a whole number (no remainder), it’s a factor. You can also use divisibility rules to eliminate some possibilities quickly. For example, if a number is not divisible by 2 or 5 it cannot be a factor of 60.
