What Are the Factors and Factor Pairs of 21?
The concept of factors of 21 is fundamental in mathematics. Whether for school exams, homework, or understanding divisibility, learning how to find the factors of 21 makes calculation and number manipulation much easier. This topic is also useful for topics like HCF, LCM, and prime factorization.
What Are Factors of 21?
A factor of 21 is a number that divides 21 exactly with no remainder. In simple words, the factors of 21 are all whole numbers that can be multiplied by another whole number to result in 21. You’ll find this concept applied in prime factorization, factor trees, and divisibility tests.
List of Positive and Negative Factors of 21
The factors of 21 include both positive and negative numbers, since a positive times a negative (or negative times a positive) also gives 21. Let’s look at all factors:
| Positive Factors | Negative Factors |
|---|---|
| 1, 3, 7, 21 | -1, -3, -7, -21 |
Factor Pairs of 21
Factor pairs are two numbers that multiply to give 21. Here are all the possible pairs:
| Pair 1 | Pair 2 | Product |
|---|---|---|
| 1 | 21 | 1 × 21 = 21 |
| 3 | 7 | 3 × 7 = 21 |
| -1 | -21 | -1 × -21 = 21 |
| -3 | -7 | -3 × -7 = 21 |
Prime Factorization of 21
Prime factorization is the process of expressing 21 as a product of its prime factors. This method is widely used for finding HCF and LCM. Here’s how to break down the number:
1. Start with 21. What is the smallest prime number that divides 21? It’s 3.2. Divide 21 by 3: 21 ÷ 3 = 7.
3. Next, check 7. 7 is a prime number.
4. So, the prime factorization is: 21 = 3 × 7
This method is called factor tree. Both 3 and 7 are prime factors. The prime factors of 21 are 3 and 7.
Is 21 a Prime or Composite Number?
A prime number has only two factors: 1 and itself. But 21 has four positive factors (1, 3, 7, and 21). That means 21 is a composite number. It can be divided by numbers other than 1 and 21. So, when asked "Is 21 a prime number?" — the answer is NO, it is composite.
How to Find Factors of 21 (Stepwise Method)
Follow these steps to find factors of 21:
1. Start with 1. 21 ÷ 1 = 21 (so 1 and 21 are factors).2. Try 2. 21 ÷ 2 = 10.5 (not a whole number, so skip).
3. Try 3. 21 ÷ 3 = 7 (so 3 and 7 are factors).
4. Try 4, 5, 6. None give a whole number result.
5. Stop at 7 (since factors start repeating in reverse).
So, the factors of 21 are 1, 3, 7, 21.
Speed Trick: Spotting Factors Fast!
Quick Trick: If the sum of the digits is a multiple of 3, then 21 is divisible by 3 (since 2 + 1 = 3). This trick helps you find factors of 21 quickly during exams!
Relation to Other Maths Concepts
The idea of factors of 21 is useful in finding HCF (LCM and HCF), simplifying fractions, as well as solving word problems involving grouping or divisibility. Learning factorization also builds the base for studying prime factorization and multiples.
Practice Problems: Try These Yourself!
- List all factors of 21 in pairs.
- Is 21 an even number?
- Is 21 a perfect square?
- Find common factors between 21 and 35.
- What are the prime factors of 21?
Common Mistakes and Tips
- Missing factor pairs by stopping checks too early (stop at √21, not 21).
- Confusing multiples with factors. Remember: factors divide into 21, multiples come from multiplying 21.
- Forgetting negative factors — for exams that ask for all integer factors, include negatives!
Classroom Tip
To quickly spot factors of 21, remember: 1 and 21 are always included, and check all numbers up to √21. Pair those that "fit" as factor pairs. At Vedantu, teachers encourage students to draw quick factor trees for better memory and visualization.
We explored factors of 21—from definition, list, prime factorization, pairs, calculation steps, and quick-check tricks. Keep practicing with Vedantu’s tips and live examples to become a pro at finding factors for any number!
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FAQs on Factors of 21 Explained for Students
1. What are the factors of 21?
The factors of 21 are the whole numbers that divide 21 without leaving a remainder. These are 1, 3, 7, and 21. The positive factors are 1, 3, 7, and 21. Including negative factors, the complete list is -21, -7, -3, -1, 1, 3, 7, 21.
2. How many factors does 21 have?
21 has four positive factors (1, 3, 7, 21). If we include negative factors, it has a total of eight factors (-21, -7, -3, -1, 1, 3, 7, 21).
3. Is 21 a prime number?
No, 21 is not a prime number. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself. 21 is divisible by 1, 3, 7, and 21, making it a composite number.
4. What is the prime factorization of 21?
The prime factorization of 21 is 3 x 7. This means 21 can be expressed as the product of the prime numbers 3 and 7.
5. What are the factor pairs of 21?
The factor pairs of 21 are pairs of numbers that multiply to give 21. These are (1, 21) and (3, 7).
6. How do I find the factors of a number?
To find the factors of a number, systematically divide the number by each whole number starting from 1, up to the square root of the number. If the division results in a whole number quotient, both the divisor and the quotient are factors. For example, with 21: 21 ÷ 1 = 21, 21 ÷ 3 = 7. Therefore, 1, 3, 7, and 21 are factors.
7. What is a factor tree, and how do I create one for 21?
A factor tree is a diagram showing the prime factorization of a number. For 21:
- Start with 21.
- Find a pair of factors (e.g., 3 and 7).
- Since 3 and 7 are both prime numbers, the tree is complete.
8. How are factors of 21 used in finding the Highest Common Factor (HCF)?
The HCF of two numbers is the largest number that divides both numbers without leaving a remainder. To find the HCF using factors, list the factors of each number and identify the largest common factor. For example, to find the HCF of 21 and 42, list the factors: 21 (1, 3, 7, 21); 42 (1, 2, 3, 6, 7, 14, 21, 42). The HCF is 21.
9. How are factors of 21 used in finding the Least Common Multiple (LCM)?
The LCM is the smallest number that is a multiple of both numbers. One method to find the LCM uses prime factorization. Find the prime factorization of each number and take the highest power of each prime factor present in either factorization. Multiply these highest powers together to get the LCM. For example, for 21 (3 x 7) and 6 (2 x 3), the LCM is 2 x 3 x 7 = 42.
10. What are some real-world applications of finding factors?
Finding factors is useful in many areas, including:
- Dividing quantities equally: Sharing 21 items among friends.
- Simplifying fractions: Reducing 21/42 to 1/2.
- Solving algebraic equations: Factoring quadratic expressions.
- Finding the HCF and LCM: Essential in various mathematical problems.
11. Can 21 have negative factors?
Yes, 21 can have negative factors. For every positive factor, there is a corresponding negative factor. The negative factors of 21 are -1, -3, -7, and -21.
