Factors of 21 | Prime Factorization of 21 - Explained Simply
Today we are going to present here Factor Tree of 21. The factor is the number that divides the original number. The factors of 21 are 1, 3, 7 and 21 itself.
What are the factors of 21?
The factors of 21 are all the whole numbers that, when divided into 21, result in a whole number with no remainder. We can also think of them as the whole numbers that multiply together to give 21.
Factor Tree Method of 21: Explained Simply
Let's systematically discover all the factors of 21 using a simple method called trial division. We'll start with the smallest possible factor (1) and work our way up.
Here's how to find the factors of 21:
Start with 1:
Is 1 a factor of 21?
Yes, always!
1 multiplied by 21 equals 21.
Factor Pair: 1*21 = 1 × 21 = 21 (1, 21)
Try 2:
Is 21 divisible by 2?
No. 21 is an odd number, so it cannot be divided evenly by 2 (it would leave a remainder of 1).
Result: 2 is NOT a factor.
Try 3:
Is 21 divisible by 3?
Yes! (A quick trick: add the digits of 21: 2 + 1 = 3. Since 3 is divisible by 3, 21 is too).
3 multiplied by 7 equals 21.
Factor Pair: 3*7 = 3 × 7 = 21 (3, 7)
Try 4:
Is 21 divisible by 4?
No. (4 × 5 = 20, 4 × 6 = 24. 21 is not in the 4 times table).
Result: 4 is NOT a factor.
Try 5:
Is 21 divisible by 5?
No. Numbers divisible by 5 must end in a 0 or a 5.
Result: 5 is NOT a factor.
Try 6:
Is 21 divisible by 6?
No. For a number to be divisible by 6, it must be divisible by both 2 and 3. We already know 21 is not divisible by 2.
Result: 6 is NOT a factor.
Try 7:
Is 21 divisible by 7?
Yes! We already found this when we tested 3 (3 × 7 = 21).
At this point, we've reached a number (7) that is part of a factor pair we've already discovered (3, 7). This means we've found all the unique factors, and we don't need to check any further numbers beyond the square root of 21 (which is approximately 4.58).
Congratulations! We have found all the factors of 21.
The factors of 21 are 1, 3, 7 and 21.
Pair Factors of 21
- 1 × 21 = 21 (1, 21)
- 3 × 7 = 21 (3, 7)
- 7 × 3 = 21 (7, 3)
- 21 × 1 = 21 (21, 1)
Properties of 21
- Composite Number: 21 is a composite number because it has more than two factors (it has four factors: 1, 3, 7, 21). A prime number, in contrast, has exactly two factors (1 and itself).
- Prime Factorization: The prime factors of 21 are 3 and 7. This means you can write 21 as a product of only prime numbers: 3 × 7 = 21. This is the unique prime factorization of 21.
- Proper Factors: These are all the factors of a number except the number itself. For 21, the proper factors are 1, 3, and 7.
Prime Factorization of 21
The prime factorization is the product of all the prime divisors on the left side.
Let's apply these methods to the number 21.
Using the Factor Tree Method for 21:
Start with 21:
```
21
/ \
? ?
```
Find two factors of 21: What two numbers multiply to give 21?
We know that 3 × 7 = 21.
So, we branch out to 3 and 7.
```
21
/ \
3 7
```
Check if factors are prime:
Is 3 a prime number? Yes, its only factors are 1 and 3. Circle it.
Is 7 a prime number? Yes, its only factors are 1 and 7. Circle it.
```
21
/ \
③ ⑦
```
All branches end in prime numbers. The process is complete.
The prime factorization of 21 is the product of the circled primes: 3 × 7.
Using the Division Method for 21:
Start with 21.
Divide by the smallest prime number that divides 21 evenly. The smallest prime is 2, but 21 is not divisible by 2. The next smallest prime is 3.
21 ÷ 3 = 7
```
3 | 21
|7
```
Continue with the new result (7). What is the smallest prime number that divides 7 evenly?
7 is not divisible by 2 or 3.
7 is divisible by 7 (itself).
7 ÷ 7 = 1
```
3 | 21
7 | 7
|1
```
The result is 1. The process is complete.
The prime factors are the numbers on the left: 3 and 7.
Therefore, the prime factorization of 21 is 3 × 7.
Both methods yield the same unique result, which is a fundamental principle called the Fundamental Theorem of Arithmetic: Every integer greater than 1 is either a prime number itself or can be represented as a unique product of prime numbers (ignoring the order of the factors).
Importance of Factor 21
Understanding factors isn't just a math exercise; it's a fundamental skill that underpins many mathematical concepts and even real-world applications.
Here's why factors are important:
1. Simplifying Fractions:
Factors are essential for simplifying fractions to their lowest terms.
To simplify the fraction 7/21, you find common factors for both 7 and 21. Both are divisible by 7.
(7 ÷ 7) / (21 ÷ 7) = 1/3.
2. Finding Common Denominators (LCM):
When adding or subtracting fractions, you often need a common denominator. The Least Common Multiple (LCM) of two numbers is found using their prime factors.
To add 1/3 and 1/7, you'd find the LCM of 3 and 7, which is 21 (since 3 and 7 are prime factors of 21).
3. Understanding Number Properties:
Factors help us classify numbers as prime or composite, perfect numbers, abundant numbers, etc. This deepens our understanding of number theory.
4. Algebra and Factoring Expressions:
In algebra, factoring is a crucial technique for solving equations and simplifying expressions. This builds directly on the concept of numerical factors.
Factoring the expression 3x + 21 involves finding the common factor of 3 and 21, which is 3. So, 3x + 21 = 3(x + 7).
5. Real-World Grouping and Division:
Whenever you need to divide a group of items into equal smaller groups, factors come into play.
If you have 21 students and want to divide them into equal teams, you could have 3 teams of 7 students or 7 teams of 3 students. You can't make 4 teams of students without having some left over.