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Prime Factors are those prime numbers that divide a number completely without leaving any remainder. Prime Numbers are those numbers that have only two factors which are 1 and the number itself. For example, the number 8 has three factors 1, 2, and 8. Here 2 is the only prime factor of 8 as it divides 8 completely without leaving a remainder and is a prime number too. 1 is neither composite nor prime and 8 is a composite number.
Read More: NCERT Solutions for Class 8 Mathematics Squares and Square Roots
Key Terms: Prime Factors,Prime Factorisation, Prime Number, Composite Number, Factor Tree, Factors, Division, Quotient
What are Prime Numbers?
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Prime Numbers are numbers that are divisible by only two factors which are 1 and the number itself.
- 2 is the smallest and the only even prime number.
- The number 1 is neither a prime number nor a composite number.
- Examples of prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, etc.
Example: 13 is a Prime Number. On the prime factorization of 13, we get
13 = 1 x 13
Thus, 13 has only two factors which are 1 and 13. Therefore, it is a prime number.
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Prime Factorization
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Prime Factorization is a method of expressing a number as a product of its prime factors. To find the prime factors of a number, divide the original number by prime factors until the remainder equals 1.
| Example: Find the Prime Factorization of 40. To perform the prime factorization of 40, we need to follow the given steps: Step 1: Divide the number 40 by the smallest prime number such that the prime number should divide the number completely. 40 ÷ 2 = 20 Step 2: Divide the quotient obtained by the smallest prime number again. 20 ÷ 2 = 10 Step 3: Repeat the same process until the quotient becomes 1. 10 ÷ 2 = 5 Step 4: Now, the number 5 is not divisible by 2, so we will move on with the next prime number which is 3. 3 also does not divide 5, so we will proceed with 5 which is the next prime number. 5 ÷ 5 = 1 Since we get 1 as the quotient, we will not proceed. Thus, the Prime Factorization of 40 is 2 × 2 × 2 × 5. |
Read More: Finding Square Roots through Prime Factorization
How to Check Whether a Number is Prime or Not?
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There are two methods that are used to determine whether a given number is a prime or not.
Method 1
2 is the only even prime number. 2 and 3 are the only two consecutive natural numbers that are prime numbers. Aside from those, every prime number can be written as 6n + 1 or 6n - 1 (except prime number multiples, i.e. 2, 3, 5, 7, 11), where n is a natural number.
For example:
- 6(1) – 1 = 5
- 6(1) + 1 = 7
- 6(2) – 1 = 11
- 6(2) + 1 = 13
- 6(3) – 1 = 17
- 6(3) + 1 = 19
- 6(4) – 1 = 23
- 6(4) + 1 = 25 (multiple of 5)
Method 2
In order to find the prime numbers greater than 40, we need to use the given formula
(n2 + n + 41), where n = 0, 1, 2,.... 39
For example:
- 02 + 0 + 41 = 41
- 12 + 1 + 41 = 43
- 22 + 2 + 41 = 47
Read More: Squares and Square Roots MCQs
Prime Factors List
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The prime factors of numbers up to 100 are as follows:
| Number | Prime Factors | Number | Prime Factors |
|---|---|---|---|
| 2 | 2 | 51 | 3, 17 |
| 3 | 3 | 52 | 2, 2, 13 |
| 4 | 2, 2 | 53 | 53 |
| 5 | 5 | 54 | 2, 3, 3, 3 |
| 6 | 2, 3 | 55 | 5, 11 |
| 7 | 7 | 56 | 2, 3, 7 |
| 8 | 2, 2, 2 | 57 | 3, 19 |
| 9 | 3, 3 | 58 | 2, 29 |
| 10 | 2, 5 | 59 | 59 |
| 11 | 11 | 60 | 2, 2, 3, 5 |
| 12 | 2,2, 3 | 61 | 61 |
| 13 | 13 | 62 | 2, 31 |
| 14 | 2, 7 | 63 | 3, 3, 7 |
| 15 | 3, 5 | 64 | 2, 2, 2, 2, 2, 2 |
| 16 | 2, 2, 2, 2 | 65 | 5, 13 |
| 17 | 17 | 66 | 2, 3, 11 |
| 18 | 2, 3, 3 | 67 | 67 |
| 19 | 19 | 68 | 2, 2, 17 |
| 20 | 2, 2, 5 | 69 | 3, 23 |
| 21 | 3, 7 | 70 | 2, 5, 7 |
| 22 | 2, 11 | 71 | 71 |
| 23 | 23 | 72 | 2, 2, 2, 3, 3 |
| 24 | 2, 2, 2, 3 | 73 | 73 |
| 25 | 5, 5 | 74 | 2, 37 |
| 26 | 2, 13 | 75 | 3, 5, 5 |
| 27 | 3, 3, 3 | 76 | 2, 2, 19 |
| 28 | 2, 2, 7 | 77 | 7, 11 |
| 29 | 29 | 78 | 2, 3, 13 |
| 30 | 2, 3, 5 | 79 | 79 |
| 31 | 31 | 80 | 2, 2, 2, 2, 5 |
| 32 | 2, 2, 2, 2, 2 | 81 | 34 |
| 33 | 3, 11 | 82 | 2, 41 |
| 34 | 2, 17 | 83 | 83 |
| 35 | 5, 7 | 84 | 2, 2, 3, 7 |
| 36 | 2, 2, 3, 3 | 85 | 5, 17 |
| 37 | 37 | 86 | 2.43 |
| 38 | 2, 19 | 87 | 3, 29 |
| 39 | 3,13 | 88 | 2, 2, 2, 11 |
| 40 | 2, 2, 2, 5 | 89 | 89 |
| 41 | 41 | 90 | 2, 3, 3, 5 |
| 42 | 2, 3, 7 | 91 | 7, 13 |
| 43 | 43 | 92 | 2, 2, 23 |
| 44 | 2, 2, 11 | 93 | 3, 31 |
| 45 | 3, 3, 5 | 94 | 2, 47 |
| 46 | 2, 23 | 95 | 5, 19 |
| 47 | 47 | 96 | 2, 2, 2, 2, 2, 3 |
| 48 | 2, 2, 2, 2, 3 | 97 | 97 |
| 49 | 7, 7 | 98 | 2, 7, 7 |
| 50 | 2, 5, 5 | 99 | 3, 3, 11 |
| 100 | 2, 2, 5, 5 | ||
Read More: Square Root 1 to 100
How to Find Prime Factors of a Number?
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The prime factors of a number can be found with the help of two methods:
- Prime Factors by Division Method
- Prime Factors using Factor Tree
Prime Factors Using Factor Tree Method
To find the prime factors of a number using the factor tree method, take the number and then divide it into any of the factors using two arrows to symbolize the branches of a tree. This procedure is repeated until the prime factors are obtained.
Example: Using the Factor Tree approach, determine the prime factors of 42.
It is clear that 2, 3, and 7 are the prime factors of 42. Thus, the prime factorization of 42 is as follows:
42 = 2 x 3 x 7
The fact that the factor tree depicts the breakdown into factors graphically is one advantage of employing this approach.
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| Related Topics | ||
|---|---|---|
| Square Roots | Number System | Square Root Formula |
| Number Theory | Square Numbers | Estimating Square Root |
Prime Factors Using Division Method
In the division method, begin by dividing the number by its smallest prime factor and continue dividing the resultant quotient by its smallest prime factor until we reach 1 as the final quotient.
Example: Using the Repeated Division Method, find the prime factors of 42.
In this case, divide 42 by the prime factor 2 to get the quotient 21, which is then divided by 3. The division is continued until we get 1 as the quotient.
As a result, we can conclude that the prime factors of number 42 are 2, 3, and 7.
Things to Remember
- Prime Factor is a number that is both a prime number and a factor of any given number.
- Prime numbers are numbers with only two factors namely 1 and the number itself.
- Prime Factorization is the method of finding prime factors of a number.
- Division Method and Factor Tree are the two methods to find the prime factors of a number.
- 2 is the only even prime factor of any number.
Sample Questions
Ques. Find the prime factors of 13. (1 Mark)
Ans. 13 is a prime number, so it can only be divided by two numbers, which are 1 and the number itself. Hence, the prime factors of 13 are 1 and 13.
Ques. Check whether 97 is a prime number or not. (2 Marks)
Ans. When we perform the prime factorization of 97, we note that the number 97 has only two factors 1 and 97. Since 97 is only divisible by 1 and itself, it is a prime number.
Ques. How many prime numbers are there between 1 to 100? (2 Marks)
Ans. There are 25 prime numbers between 1 to 100. The prime numbers between 1 to 100 are 2,3,5,7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59,61, 67, 71,73, 79, 83, 89, and 97. Here, 2 is the only even prime number.
Ques. Find the prime factors for 60. (2 Marks)
Ans. To find the prime factors of 60, we will follow the given steps:
Divide 60 by 2
60 ÷ 2 = 30
Divide 30 by 2 again to get 15
30 ÷ 2 = 15
Now, 15 is divisible by 3 followed by 5.
15 ÷ 3 = 5
5 ÷ 5 = 1
The prime factorization of 60 is 2 x 2 x 3 x 5. Thus, the prime factors of 60 are 2, 3, and 5.
Ques. Find the prime factors of 12. (2 Marks)
Ans. Divide 12 by the smallest prime number 2.
12 ÷ 2 = 6
6 is not a prime number, so continue the division.
6 ÷ 2 = 3
3 is a prime number, so it will be divisible by 3 only.
3 ÷ 3 = 1
Thus, the prime factorization of 12 is 2 x 2 x 3 with 2 and 3 as its prime factors.
Ques. Find the prime factors of 12 and 36. (2 Marks)
Ans. The prime factors of 12 and 36 are
- 12 = 2 x 2 x 3.
- 36 = 2 x 2 x 3 x 3
Thus, the prime factors of both 12 and 36 are 2 and 3.
Ques. Find the prime factors of 7084. (2 Marks)
Ans. Factors of 7084 can be determined as follows:
- 7084 = 3542 × 2
- 3542 = 1771 × 2
- 1771 = 253 × 7
- 253 = 23 × 11
Hence, the prime factors are 2, 2, 7, 11, 23.
Ques. List the common prime factors of 152 and 76. (2 Marks)
Ans. We will first find the prime factorization of 152 and 76,
- Prime Factorization of 152 = 2 × 2 × 2 × 19
- Prime Factorization of 76 = 2 × 2 × 19
Thus, the common prime factors of 152 and 76 are 2 and 19.
Ques. Find the prime factors of 278. (2 Marks)
Ans. Using the division approach:
- 278 ÷ 2 = 142.
- 142 ÷ 2 = 71
- 71 ÷ 71 = 1.
Thus, the prime factors of 278 are 2 and 71.
Ques. Determine the prime factors of 50034. (2 Marks)
Ans. The prime factors of 50034 are
- 50034 = 25017 × 2
- 25017 = 8339 × 3
- 8339 = 269 × 31
Therefore, the Prime Factors of 23356 are 2, 3, 31, and 269.
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