The prime factors of a number are the 'prime numbers' that are multiplied to get the original number. For example, 2 and 5 are the prime factors of 20, i.e., 2 × 2 × 5 = 20. We know that the 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. Therefore, it should be noted that all the factors of a number may not necessarily be prime factors.
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.
Prime factorization of a number:
| 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 |
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 by factor tree method
- Prime factorization by division method
(i) Factor Tree Method of Prime Factorization
In the factor tree approach, the factors of a given number are obtained in pairs and then those numerals obtained are further factorized until the pair factors are prime numbers. The steps to estimate the prime factorization of a number utilising the factor tree method are as follows:
- Step 1: Position the number for which the factors are to be calculated on top of the factor tree.
- Step 2: Start writing the factor of the number in pairs, like the branches of the tree.
- Step 3: Next, factorise the composite factors obtained in the previous step and form the next branches of the tree.
- Step 4: Keep on repeating the same process until we reach a pair factor with both prime numbers.
Given below are two examples to understand the same. The tree below is the factor tree for the number 42.
In the first branch, 42 is divided into pairs of 3 and 14 as 3 × 14 = 42.
Next, 3 is a prime number hence is kept as it is and 14 is further factorised as 2 and 7 as 2 × 7 = 14.
Finally, we are having a pair of 2 and 7 where both the numbers are prime, thus we stop drawing further branches.
Thus, 42 as a result of the prime factorisation is; 3 × 2 × 7= 42.
Next, we have factor tree of 36.
Here, 36 is divided into factor pairs of 2 and 18, as 2 × 18 = 36.
Next, 18 being a composite number is further factorized into a branch of 2 and 9.
Again, making the branch of 9, we get the pair as 3 and 3. Here as both the numbers are prime, we complete the process of making branches.
Thus, 36 as a result of the prime factorisation is; 2 × 2 × 3 × 3 = 22×33 = 36.
(ii) Division Method of Prime Factorization
Similar to the factor tree, the division method can also be used to obtain the different prime factors of the number. The steps for the prime factorization by division method are as follows.
- Step 1: Depending on the number given, start dividing it by the smallest prime number(commonly 2 for the even number and 3 for the odd number)such that the number is completely divisible by the number.
- Step 2: In the next step, divide the quotient of the previous step again by the least prime number.
- Step 3: Continue this process, until we have the quotient as 1.
- Step 4: Lastly, take the product of all the prime factors that are present in the divisors column.
Given below are two examples to understand the same using Divisibility Rules. In the below example, the prime factors of a number i.e. 60 are calculated.
Here we first divide the number 60 by 2; 60 ÷ 2 = 30.
Next, 30 i.e. the quotient is again divided by 2; 30 ÷ 2 = 15.
Now the quotient left is 15 and it is not divisible by 2. Thus we will move to the next prime number i.e. 3. Thus 15 ÷ 3 = 5.
Lastly, diving 5 by 5 we get the remainder as 1.
60 as the product of prime factors is; 22 × 3 × 5 =60.
The prime factorization example by the division method for the number 28 is as follows.
Here also we first start to divide the number 28 by 2; 28 ÷ 2 = 14.
Next, 14 i.e. the quotient is again divided by 2; 14 ÷ 2 = 7.
Now 7 being a prime number is only divisible by 7 and we have the quotient as 1.
28 as the product of prime factors is; 22 × 7 = 28.
