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Factors of 21 are the natural numbers that divide the number 21 completely leaving zero as the remainder. The number 21 is an odd composite number which means it has more than two factors. It has a total of 4 factors.
- 1, 3, 7, and 21 are the factors of 21.
- 1 is the smallest factor of 21 while 21 itself is the biggest factor.
- The prime factorization of 21 is 3 × 7 with 3 and 7 being the two prime factors of 21.
- Factor pairs of 21 are the numbers resulting in the original number when multiplied.
- The positive factor pairs of 21 are (1, 21) and (3, 7).
- The negative pair factors of 21 are (-1, -21) and (-3, -7).
- The sum of the factors of 21 is 32.
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Key Terms: Factors of 21, Prime Factorization, Factor Pairs, Remainder, Composite Number, Prime Factors, Pair Factors, Multiplication
What are Factors of 21?
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Factors of 21 are the numbers that divide 98 evenly with no remainder. It means that the numbers that leave zero as the remainder on division with 21 are the factors of 21. 21 is a composite number, thus, it has more than two factors. The factors of 21 are 1, 3, 7, and 21.- Factors of 21: 1, 3, 7 and 21
- Negative Factors of 21: -1, -3, -7 and -21
- Prime Factorization of 21: 3 × 7 = 3 × 7
- Prime Factors of 21: 3, 7
- Sum of Factors of 21: 32
Factors of 21
How to Calculate Factors of 21?
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In order to find the factors of 21, the following steps need to be followed:
- Step 1: Write down the number 21 as the first step.
- Step 2: Find out any two numbers, which result in 21 on multiplication. Take 3 and 7, such that 3 × 7 = 21.
- Step 3: 3 and 7 are prime numbers with only two factors. Thus, we cannot factorize them further.
- Step 4: Thus, the factorization of 21 is expressed as 21 = 3 × 7 × 1.
Thus, there are four factors of 21 which are 1, 3, 7, and 21.
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Factors of 21 in Pairs
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Factors Pairs of 21 are a set of two numbers that give the original number 21 on multiplication with each other. The pair factors of 21 can be positive or negative integers but not fractions or decimals.
- Positive Pair Factors of 21: (1, 21) and (3, 7).
- Negative Pair Factors of 21: (-1, -21) and (-3, -7).
Both positive and negative pair factors of 21 are listed below.
Positive Pair Factors of 21
The positive pair factors of 21 are:
| Positive Factors of 21 | Positive Pair Factors of 21 |
|---|---|
| 1 × 21 = 21 | (1, 42) |
| 3 x 7 = 21 | (3, 7) |
Negative Pair Factors of 21
The factor pairs of 21 can be negative too as the product of two negative numbers is positive, thus, the negative pair factors of 21 are as follows:
| Negative Factors of 21 | Negative Pair Factors of 21 |
|---|---|
| (-1) × (-21) = 21 | (-1, -21) |
| (-3) × (-7) = 21 | (-3, -7) |
Factors of 21 by Prime Factorization
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Prime Factorization refers to the process of expressing a composite number as the product of its prime factors. The factors of 21 can be calculated using the prime factorization method. For the prime factorization of 21, follow the given steps:
Step 1: First, divide 21 by its prime factor, for instance, 3.
| 21 ÷ 3 = 7 |
Step 2: Now, divide 7 by its prime factor until the quotient is obtained as 1.
| 7 ÷ 7 = 1 |
We have received 1 at the end, so, we can not proceed with division further. Thus, the prime factorization of 21 is 3 x 7, where the prime factors of 21 are 3 and 7.
Factor Tree of 21
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The factors of 21 can also be represented with the help of the factor tree. The end of the branches of the factor tree has the prime factors of a number that cannot be factorized further.
The factor tree of 21 is as follows:
Solved Examples on Factors of 21
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Here are some solved examples on Factors of 21:
Example 1: What are the common factors of 21 and 18?
Solution: The factors of 21 and 18 are:
- Factors of 21: 1, 3, 7, 21
- Factors of 18: 1, 2, 3, 6, 9, 18
Thus, 1 and 3 are the common factors of 21 and 18.
Example 2: What is the Find the Least Common Multiple (LCM) and Greatest Common Factor (GCF) of 21 and 11?
Solution: First we will write down all the factors of 21 and 11:
- Factors of 21: 1, 3, 7, 21
- Factors of 11: 1, 11
Thus, the Least Common Multiple (LCM) of 21 and 11 is 231. The Greatest Common Factor (GCF) of 21 and 11 is 1.
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Things to Remember
- Factors of 21 are the numbers that divide 21 completely with zero as the remainder.
- 21 is a composite number, thus, it has four factors in total.
- 1, 3, 7, and 21 are the factors of 21.
- The sum of all the factors of 21 is equal to 32.
- Factor Pairs of 21 are those numbers that give the original number 21 when multiplied in pairs.
- The factors of 21 in pairs are (1, 21) and (3, 7).
- The prime factorization of 21 is 3 x 7.
- 3 and 7 are the two prime factors of 21.
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Sample Questions
Ques. What are the factors of 21? (2 Marks)
Ans. The factors of 21 are as follows:
- 21 ÷ 1 = 21
- 21 ÷ 3 = 7
- 21 ÷ 7 = 3
- 21 ÷ 21 = 1
Thus, the factors of 21 are 1, 3, 7, and 21.
Ques. Write down the factors and multiples of 21. (2 Marks)
Ans. The factors and multiples of 21 are as follows:
- Multiples of 21: 21, 42, 63, 84, 105, 126, 147, 168, 189, 210, etc.
- Factors of 21: 1, 3, 7, 21.
Ques. Find out the Greatest Common Factor of 21 and 15. (2 Marks)
Ans. First, we will write down all the factors of 21 and 15:
- Factors of 21: 1, 3, 7, 21
- Factors of 15: 1, 3, 5, 15
Thus, the common factors of 21 and 15 are 1 and 3. Therefore, the Greatest Common Factor of 21 and 15 is 3.
Ques. List the first 10 multiples of 21. (2 Marks)
Ans. The first 10 multiples of 10 are as follows:
- 21 × 1 = 21
- 21 × 2 = 42
- 21 × 3 = 63
- 21 × 4 = 84
- 21 × 5 = 105
- 21 × 6 = 126
- 21 × 7 = 147
- 21 × 8 = 168
- 21 × 9 = 189
- 21 × 10 = 210
Thus, the first 10 multiples of 21 are 21, 42, 63, 84, 105, 126, 147, 168, 189, and 210.
Ques. What are the common factors of 21 and 24? (2 Marks)
Ans. Writing down all the factors of 21 and 24,
- Factors of 21: 1, 3, 7, and 21.
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, and 24.
Thus, there are 2 common factors of 21 and 24 which are 1 and 3.
Ques. Does 21 have 2 factors only? (2 Marks)
Ans. No, the number 21 has 2 more factors other than 1 and 21 itself. Since 21 is an odd composite number, it has more than 2 factors which are 1, 3, 7, and 21.
- 21 ÷ 1 = 21
- 21 ÷ 3 = 7
Thus, 1, 3, 7, and 21 itself are the factors of 21.
Ques. Three girls bought 21 chocolates from a shop and distributed them among themselves equally. How many chocolates will each one of them get? (2 Marks)
Ans. There is a total of 21 chocolates that are to be divided among 3 girls equally.
It means that we need to divide 21 by 3
21 ÷ 3 = 7
Thus, each of the girls will get 7 candies.
Ques. What will be the sum of the factors of 21? (2 Marks)
Ans. We know that the factors of 21 are 1, 3, 7, and 21.
Adding up the factors, we get
1 + 3 + 7 + 21 = 32
Thus, the sum of the factors of 21 is 32.
Ques. What is the sum of all prime factors of 21? (2 Marks)
Ans. Out of all the factors 21 which are 1, 3, 7, and 21, there are only 2 prime factors which are 3 and 7.
Thus, adding up both prime factors of 21, we get
3 + 7 = 10
So, the sum of prime factors of 21 is 10.
Ques. Is 7 a factor of 21 or not? Explain. (2 Marks)
Ans. In order to check whether 7 is a factor of 21 or not, we need to divide 21 by 7 and check whether the remainder is zero or not.
21 ÷ 7 = 3
It can be seen that there is no remainder left at the end of the division process, thus, 7 is a factor of 21.
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