Prime Factorization
Prime factorization is a way of expressing a number as a product of its prime factors. A prime number is a number that has exactly two factors, 1 and the number itself. For example, if we take the number 30. We know that 30 = 5 × 6, but 6 is not a prime number. The number 6 can further be factorized as 2 × 3, where 2 and 3 are prime numbers. Therefore, the prime factorization of 30 = 2 × 3 × 5, where all the factors are prime numbers.
Let us learn more about prime factorization with various mathematical problems followed by solved examples and practice questions.
| 1. | What is Prime Factorization? |
| 2. | Prime Factorization of a Number |
| 3. | Methods of Prime Factorization |
| 4. | FAQs on Prime Factorization |
What is Prime Factorization?
The process of writing a number as the product of prime numbers is prime factorization. Prime numbers are the numbers that have only two factors, 1 and the number itself. For example, 2, 3, 5, 7, 11, 13, 17, 19, and so on are prime numbers.
Prime factorization of any number means to represent that number as a product of prime numbers. For example, the prime factorization of 40 can be done in the following way:
Prime Factorization of a Number
Let us see the prime factorization chart of a few more numbers in the table given below:
| Numbers | Prime Factorization |
|---|---|
| 36 | 22 × 32 |
| 24 | 23 × 3 |
| 60 | 22 × 3 × 5 |
| 18 | 2 × 32 |
| 72 | 23 × 32 |
| 45 | 32 × 5 |
| 40 | 23 × 5 |
| 50 | 2 × 52 |
| 48 | 24 × 3 |
| 30 | 2 × 3 × 5 |
| 42 | 2 × 3 × 7 |
What are Factors and Prime Factors?
Factors of a number are the numbers that are multiplied to get the original number. For example, 4 and 5 are the factors of 20, i.e., 4 × 5 = 20, whereas, prime factors of a number are the prime numbers that are multiplied to get the original number. For example: 2, 2, and 5 are the prime factors of 20, i.e., 2 × 2 × 5 = 20.
Prime factorization is similar to factoring a number but it considers only prime numbers (2, 3, 5, 7, 11, 13, 17, 19, and so on) as its factors. Therefore, it can be said that factors that divide the original number completely and cannot be split into more factors are known as the prime factors of the given number.
Methods of Prime Factorization
There are various methods for the prime factorization of a number. The most common methods that are used for prime factorization are given below:
- Prime factorization using factor tree method
- Prime factorization using division method
Prime Factorization using Factor Tree Method
In the factor tree method, the factors of a number are found and then those numbers are further factorized until we reach the prime numbers. To evaluate the prime factorization of a number using the factor tree method, we use the steps given below:
- Step 1: Place the number on top of the factor tree.
- Step 2: Then, write down the corresponding pair of factors as the branches of the tree.
- Step 3: Factorize the composite factors that are found in step 2, and write down the pair of factors as the next branches of the tree.
- Step 4: Repeat step 3, until we get the prime factors of all the composite factors.
Example: Follow the figure given below to understand the concept and do the prime factorization of 850.
Prime Factorization using Division Method
The division method can also be used to find the prime factors of a large number by dividing the number by prime numbers. Follow the steps given below to find the prime factors of a number by using the division method:
- Step 1: Divide the number by the smallest prime number such that the smallest prime number should divide the number completely.
- Step 2: Again, divide the quotient of step 1 by the smallest prime number.
- Step 3: Repeat step 2, until the quotient becomes 1.
- Step 4: Finally, multiply all the prime factors that are the divisors.
Example: Let us do the prime factorization of 60 with the division method using the steps given above.
Prime factorization of 60 = 2 × 2 × 3 × 5
Therefore, the prime factors of 60 are 2, 3, and 5.
Applications of Prime Factorization
Prime factorization is used extensively in the real world. The two most important applications of prime factorization are given below.
- Cryptography and Prime Factorization
- HCF and LCM Using Prime Factorization
Cryptography and Prime Factorization
Cryptography is a method of protecting information using codes. Prime factorization plays an important role for the coders who create a unique code using numbers which is not too heavy for computers to store or process quickly.
HCF and LCM Using Prime Factorization
To find the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of two numbers, we use the prime factorization method. For this, we first do the prime factorization of both the numbers. The following points related to HCF and LCM need to be kept in mind:
- HCF is the product of the smallest power of each common prime factor.
- LCM is the product of the greatest power of each common prime factor.
Example: What is the HCF and LCM of 850 and 680?
Solution: We will first do the prime factorization of both the numbers. The prime factorization of 850 is: 850 = 2 × 52 × 17. The prime factorization of 680 is: 680 = 23 × 5 × 17
HCF is the product of the smallest power of each common prime factor. Hence, HCF of (850, 680) = 21 × 51 × 171 = 170. LCM is the product of the greatest power of each common prime factor. Hence, LCM of (850, 680) = 23 × 52 × 171 = 3400. Thus, HCF of (850, 680) = 170, LCM of (850, 680) = 3400
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Prime Factorization Examples
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Example 1: Express 1080 as the product of prime factors.
Solution:
We will do the prime factorization of 1080 as follows:
Thus, 1080 = 23 × 33 × 5
Therefore, the prime factorization of 1080 is 23 × 33 × 5
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Example 2: Find the lowest common multiple of 48 and 72 using prime factorization.
Solution:
We will do the prime factorization of 48 and 72 as shown below:
The prime factorization of 72 is shown below:
The LCM or lowest common multiple of any 2 numbers is the product of the greatest power of the common prime factors. Hence, LCM (48, 72) = 24 × 32 = 144
Therefore, LCM (48, 72) = 24 × 32 = 144
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Example 3: Jane needs to prove that the prime factorization of 40 will always remain the same. Can you help her to prove it?
Solution:
Jane can use the division method and the factor tree method to prove that the prime factorization of 40 will always remain the same. Jane knows that 40 can be factored as 5 and 8. The composite number 8 can further be broken down as a product of 2 and 4. The number 5 is already a prime number. Hence, she will show the division method and factor tree method in the following way:
Therefore, this shows that by any method of factorization, the prime factorization remains the same. The prime factorization for a number is unique.
FAQs on Prime Factorization
What is Prime Factorization in Math?
Prime factorization of any number means to represent that number as a product of prime numbers. A prime number is a number that has exactly two factors, 1 and the number itself.
How to do Prime Factorization?
Prime factorization of any number can be done by using two methods:
- Division method - In this method, the given number is divided by the smallest prime number which divides it completely. After this, the quotient is again divided by the smallest prime number. This step is repeated until the quotient becomes 1. Then, all the prime factors that are divisors are multiplied.
- Factor tree method - In this method, the given number is placed on top of the factor tree. Then the corresponding pairs of factors are written as the branches of the tree. After this step, the composite factors are again factorized and written down as the next branches. This procedure is repeated until we get the prime factors of all the composite factors.
What is the Prime Factorization of 72, 36, and 45?
Prime factorization is the way of writing a number as the multiple of their prime factors. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, and so on. The prime factorization of 72, 36, and 45 are shown below.
- Prime factorization of 72 = 23 × 32
- Prime factorization of 36 = 22 × 32
- Prime factorization of 45 = 32 × 5
How to Find LCM using Prime Factorization?
The abbreviation LCM stands for 'Least Common Multiple'. The Least Common Multiple (LCM) of a number is the smallest number that is the product of two or more numbers. The LCM of two numbers can be found out by first finding out the prime factors of the numbers. The LCM is the product of the greatest power of each common prime factor.
How to Find HCF using Prime Factorization?
The abbreviation HCF stands for 'Highest Common Factor'. The Highest Common Factor (HCF) of two numbers is the highest possible number which divides both the numbers completely. The HCF of two numbers can be found out by first finding out the prime factors of the numbers. The HCF is the highest common factor from the prime factors of the two numbers.
Where is Prime Factorization Useful?
Prime factorization is used to find the HCF and LCM of numbers. It is widely used in cryptography which is the method of protecting information using codes. Prime numbers are used to form or decode those codes.
What is the Prime Factorization of 24?
The number 24 can be written as 4 × 6. Now the composite numbers 4 and 6 can be further factorized as 4 = 2 × 2 and 6 = 2 × 3. Therefore, the prime factorization of 24 is 24 = 2 × 2 × 2 × 3 = 23 × 3
How is Prime Factorization used in the Real World?
Prime factorization is used extensively in the real world. For example, if we need to divide anything into equal parts, or we need to exchange money, or calculate the time while travelling, we use prime factorization. One common example is, if we have 21 candies and we need to divide it among 3 kids, we know the factors of 21 as, 21 = 3 × 7. This means we can distribute 7 candies to each kid.
When to use Prime Factorization?
Prime factorization is one of the methods used to find the Greatest Common Factor (GCF) of a given set of numbers. GCF by prime factorization is useful for larger numbers for which listing all the factors is time-consuming.