Prime factorization is expressing a positive integer as product of its prime factors. Consider, for example the number 100. The factors of 100 are: 1, 2, 4, 5, 10, 20, 50 and 100. Among the factors of 100, only 2 and 5 are prime numbers. Therefore, the prime factors of 100 are 2 and 5. If 100 is expressed as product of only the prime factors 2 and 5, then such an expression is called prime factorization of 100. Below, we will discuss two methods of carrying out prime factorization of a natural number.

They are: Factor tree method and Division method. ## 1. The factor tree method of prime factorization:

In the above method, the natural number 100 was repeatedly expressed as product of its factors, until all of the factors are prime numbers.

100 = 2 × 50 = 2 × 2 × 25 = 2 × 2 × 5 × 5

Using exponents laws, 100 = 22 × 52

This method of resolving a number into only its prime factors is called prime factorization. And this method of prime factorization is called factor tree method.

Now, let us resolve 100 into its factors differently: 100 = 50 × 2 = 25 × 2 × 2 = 5 × 5 × 2 × 2

Using exponents laws, 100 = 52 × 22

In this factor tree of prime factorization, though the order of prime factors is different, but 100 has been resolved into a same group of prime factors, i.e. 2 and 5. So, the prime factorization of a natural number using the factor tree method yields a same group of prime factors

A few examples of factor tree method of prime factorization:

210 = 2 × 105 = 2 × 3 × 35 = 2 × 3 × 5 × 7
84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7

From these examples, factor tree method is a protracted procedure, involving repetition of writing same factors. The division method proves better by being easier and faster.

## 2. Division method of prime factorization

Let us divide number 100 only by its prime factors, starting from 2 as below:

The above method is dividing a natural number, such as 100, by its prime factors. The division process ends, when 1 is the quotient, or the number to be divided further.

Therefore, the prime factorization of 100 thru the division method is

100 = 2 × 2 × 5 × 5

Using exponent laws, the prime factors can be grouped as under

100 = 22 × 52

Some more examples of prime factorization of numbers using division method:

### Problem 1:

How many prime factors does A have, if A is the product of all the positive integers from 2 through 10?

### Solution:

A = 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10 =

2 × 3 × 2 × 2 × 5 × 2 × 3 × 7 × 2 × 2 × 2 × 3 × 3 × 2 × 5 =

28 × 34 × 52 × 71

Therefore, A = 28 × 34 × 52 × 71.

But 28 × 34 × 52 × 71 is prime factorization of A.

Therefore the prime factors of A will be the bases in the product:

28 × 34 × 52 × 71, which are 2, 3, 5 and 7. Hence A has 4 prime factors.

### Problem 2:

Find the greatest prime factor in the following sum: 3 + 32 + 33

### Solution:

3 + 32 + 33 = 3 × (1 + 3 + 32) = 3 × (13) = 3 × 13

Since, 3 × 13 constitutes prime factorization of the given sum, the greatest prime factor is 13 in the sum: 3 + 32 + 33