Prime Factorization Formula

Before going to learn the prime factorization formula, let us recall what is prime factorization. It is a way of expressing a number as a product of its prime factors. The fundamental theorem of arithmetic states that "Every composite number can be factorized as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur." The prime factorization formula helps in finding the prime factorization of any number.

What Is the Prime Factorization Formula?

We can express any composite number as the product of powers of prime numbers and this way of writing the composite number as the product is called the prime factorization. The prime factorization formula of any number is given as:

N = X^a × Y^b × Z^c

where,

• N = Any number 
• X, Y, and Z = Prime factors of number N
• a, b, and c = exponents of prime factors X, Y, and Z respectively

How to find the prime factorization of a number?

The following method and formulas can be used to calculate the prime factorization of any number:

1. Division Method

The steps for calculating the prime factors of a number is similar to the process of finding the factors of any number.

• Start dividing the number by the smallest prime number i.e., 2, followed by 3, 5, and so on to find the smallest prime factor of the number. 
• Again, divide the quotient by the smallest prime number.
• Repeat the process, until the quotient becomes 1 after repeated division.
• Finally, represent the number as a product of all the prime factors.

2. Factor Tree Method

Represent the given number in form of a tree. 

• Keep the number in the centered position as root. 
• Divide the number with its smallest prime factor and represent the factor as a number in one branch.
• Represent the quotient obtained in the other branch and repeat the above point for it, until you obtained 1 as the factor for the remaining number.
• Each branch of the tree thus obtained will eventually end in a prime number. 

Let us understand the prime factorization formula in detail using examples in the following section.

Examples Using Prime Factorization Formula

Example 1: Find the prime factorization of 40 using the prime factorization formula (the division method).

Solution:     

To find: prime factorization of 40

After repeated division to obtain all prime factors of 40, we get:

Prime factorization of 40 can thus be given as:

40 = 2 × 2 × 2 × 5

Answer: Prime factorization of 40 can thus be given as: 40 = 2 × 2 × 2 × 5

Example 2: Find the prime factorization of 54 using the prime factorization formula (the factor tree method).

Solution:     

To find: prime factorization of 54

After repeated division to obtain all prime factors of 54, we get the factor tree as: