What Are the Factors and Prime Factors of 10?
The concept of factors of 10 is an essential foundation in mathematics and makes solving various problem types—ranging from basic arithmetic to advanced topics like HCF and LCM—much easier. Whether you’re tackling board exams, competitive tests, or real-life calculations, knowing the factors of common numbers like 10 helps boost speed and accuracy in problem-solving.
What Are Factors of 10?
A factor of 10 is any whole number that divides 10 exactly, leaving no remainder. For instance, since 2 × 5 = 10, both 2 and 5 are factors of 10. Similarly, 1 × 10 = 10, so 1 and 10 are also factors.
Key Formula for Factors of 10
To find factors of any number, use this process:
If \( n \) is the number, then
Factors of n = "All positive integers that divide n, i.e., n ÷ k leaves remainder 0."
For 10: Find all k such that \( 10 \div k \) is a whole number.
Step-by-Step Illustration
- Start with 1: 10 ÷ 1 = 10 (remainder 0, so 1 is a factor).Pair: 1 × 10 = 10
- Try 2: 10 ÷ 2 = 5 (remainder 0, so 2 is a factor).Pair: 2 × 5 = 10
- Try 3: 10 ÷ 3 ≈ 3.33 (not a whole number, skip).
- Try 4: 10 ÷ 4 = 2.5 (not a whole number, skip).
- Try 5: 10 ÷ 5 = 2 (remainder 0, so 5 is a factor).Pair: 5 × 2 = 10
- Try 10: 10 ÷ 10 = 1 (remainder 0, so 10 is a factor).
All Factors of 10 (List & Table)
All factors of 10 are: 1, 2, 5, and 10.
| Factor Pair | Product |
|---|---|
| (1, 10) | 1 × 10 = 10 |
| (2, 5) | 2 × 5 = 10 |
| (5, 2) | 5 × 2 = 10 |
| (10, 1) | 10 × 1 = 10 |
Prime Factorization of 10
A prime factorization breaks a number into its prime factors. For 10:
- Find the smallest prime that divides 10: 210 ÷ 2 = 5
- 5 is prime, so stop.Prime factorization is 2 × 5
So, prime factors of 10 are 2 and 5.
Factors of 10 in Negative Numbers
Factors can also be negative! For -10, the factors are -1, -2, -5, and -10, because multiplying two negatives gives a positive (e.g., (-2) × (-5) = 10).
Difference between Factors and Multiples of 10
A factor of 10 is a number that divides 10 without remainder, like 1, 2, 5, or 10. A multiple of 10 is the product of 10 and another whole number (e.g., 10, 20, 30, 40…). Factors "fit inside" a number; multiples "grow out" from it.
For a detailed explanation, click here.
Speed Trick or Vedic Shortcut
To remember the factors of numbers like 10, recall that any even number ending with '0' has 1, 2, 5, and 10 as factors. Just scan all numbers up to the square root of 10 and check divisibility.
Try These Yourself
- List all factors of 20.
- Is 3 a factor of 10? Why or why not?
- Which numbers are common factors of 10 and 20?
- What is the prime factorization of 100?
Frequent Errors and Misunderstandings
- Confusing factors with multiples (e.g., thinking 20 is a factor of 10)
- Not listing 1 or the number itself as a factor
- Missing negative factors
- Treating fractions or decimals as factors (only whole numbers count)
Relation to Other Concepts
Understanding factors of 10 lays the groundwork for related topics like HCF and LCM, prime factorization, and solving fraction questions. Mastery here will help in more advanced concepts like divisibility rules, algebraic factoring, and word problems involving multiples.
Classroom Tip
A quick memory trick for factors: If the number ends in 0, always try dividing by 2 and 5. Vedantu’s live teachers often recommend drawing a simple "factor tree" for numbers like 10: start at the top with 10, then split into 2 × 5 at the branches.
Wrapping It All Up
We explored factors of 10—definition, formula, pairs, prime factorization, common errors, and connections to other important maths ideas. Continue with Vedantu for more helpful topic guides designed by expert teachers. Practice finding factors of other numbers, such as factors of 12 or factors of 15, to build your speed and confidence.
Useful Internal Links
- Factors and Multiples – Complete guide, with definitions, tricks, and word problems.
- Multiples of 10 – Understand how multiples relate to factors.
- Factors of 100 – Practice with larger numbers.
- Factor Tree Method – Visual learning for breaking down numbers.
FAQs on Factors of 10 Explained: How to Find, List, and Use Them in Maths
1. What are the factors of 10?
The factors of 10 are the whole numbers that divide 10 without leaving a remainder. These are 1, 2, 5, and 10. They represent all the numbers that can be multiplied together to equal 10.
2. How do I find the factors of 10?
To find the factors of 10, systematically check which whole numbers divide 10 evenly. You can do this through division or by finding pairs of numbers that multiply to 10. For example: 1 x 10 = 10 and 2 x 5 = 10. Therefore, the factors are 1, 2, 5, and 10.
3. What are the factor pairs of 10?
The factor pairs of 10 are the pairs of numbers that multiply to 10. These are (1, 10) and (2, 5). Note that (5,2) and (10,1) are also considered factor pairs, but represent the same factors.
4. What is the prime factorization of 10?
The prime factorization of 10 expresses 10 as a product of its prime factors. Since 10 = 2 x 5, and both 2 and 5 are prime numbers, the prime factorization of 10 is 2 x 5.
5. Is 10 a prime or composite number?
10 is a composite number because it has more than two factors (1, 2, 5, and 10). A prime number has only two factors: 1 and itself.
6. What is the difference between factors and multiples of 10?
Factors divide a number evenly, while multiples are the results of multiplying a number by whole numbers. For example, the factors of 10 are 1, 2, 5, and 10, while some multiples of 10 are 10, 20, 30, 40, and so on.
7. Can negative numbers be factors of 10?
Yes. While we typically focus on positive factors, negative numbers can also be factors. The factors of 10 also include -1, -2, -5, and -10 because (-1) x (-10) = 10 and (-2) x (-5) = 10.
8. How are factors of 10 used to find the Highest Common Factor (HCF)?
To find the HCF of two or more numbers, list the factors of each number. The highest common factor among the sets is the HCF. For example, to find the HCF of 10 and 20, we list the factors of 10 (1,2,5,10) and the factors of 20 (1,2,4,5,10,20). The highest number that appears in both lists is 10, therefore the HCF of 10 and 20 is 10.
9. How are the factors of 10 used in finding the Lowest Common Multiple (LCM)?
The LCM is found by identifying the prime factors of each number, then finding the highest power of each prime factor present in the numbers. For example, to find the LCM of 10 and 12, the prime factorization of 10 is 2 x 5 and the prime factorization of 12 is 22 x 3. The LCM is therefore 22 x 3 x 5 = 60.
10. Why is 1 always a factor of any number?
One (1) is a factor of every number because any number divided by 1 equals itself, with no remainder. This is a fundamental property in number theory.
11. Can decimals or fractions be factors?
No. By definition, factors are always whole numbers (integers). Decimals and fractions are not considered factors in standard mathematical definitions.
12. Explain how to use a factor tree to find the factors of 10.
A factor tree visually breaks down a number into its prime factors. For 10, you would start with 10, then branch it into 2 x 5. Since 2 and 5 are both prime numbers, the tree stops here, and the prime factors are 2 and 5. All factors of 10 can be obtained from these prime factors (1, 2, 5, 10).
