Prime Factorization
2026-02-28 00:50 Diff
https://brightchamps.com/en-us/math/numbers/prime-factorization

The prime factorization can be done on any number in two different methods. They are, division method and factor tree method. We will now learn the division method.

Division method: We can find the factors of a number by dividing the given number with only prime numbers. Let us learn this method with the help of an example.

Example: Find the prime factors of 72 using the division method.

Step 1: Firstly, we have to divide 72 with its smallest prime factor, which is 2. Therefore, we get \(\frac{72}{2}=36\).

Step 2: Repeat the process until we get a number that is not any more divisible by 2. \(\frac{36}{2}=18\)

Step 3: Let us divide 18 by 2. \(\frac{18}{2}=9\)

Step 4: Since we cannot divide 9 by 2, let us divide the number with the smallest prime factor of 9. Therefore, we get \(\frac{9}{3}=3\).

Step 5: Divide the number till we get 1 as the quotient. \(\frac{3}{3}=1\)

Step 6: Now we got the prime factors of 72. It can be written as, \(72 = 2\times2\times2\times3\times3=2^3\times3^2\). Where, 2 and 3 are prime numbers and the prime factors of 72.

Factor tree method: We can also find the prime factors of any given number with the help of factor tree method. In this process, we keep on factorizing the given number until we find its prime factors. The factors are split and written in the form of branches of a tree. We circle the final factors and consider them as the prime factors of the given number. Let us understand this process by finding the prime factors of 72.

Step 1: Firstly, split 72 into two factors. Let us take 18 and 4.

Step 2: Check if these two numbers are prime or not. 

Step 3: Since the two numbers are composite, they can be divided further into more factors. Repeat this process until we get prime numbers.

Step 4: Split 4 into 2 times 2. Similarly, 18 can be split into 2 times 9. 

Step 5: We can further split 9 into 3 times 3. Now, we are left only with prime numbers. Circle all the prime numbers at the end. 

Step 6: Therefore, the prime factors of 72 are given as, \(72= 2\times2\times2\times3\times3=2^3\times3^2\)

The factor tree is not always the same. If we split 72 into 2 and 36, the form of the tree would be different. But that doesn't change the prime factors of the given number.