Prime Factorization

Prime Number:
Prime number is a number which is greater than 1 and which can be efficiently divided by 1 and by itself not by any other number.
It is also a whole number.
An Integer P > 1 is called a prime number when its only divisors are ± 1 and ± P.

Simple properties of primes:
(a) A prime 'p' is either relatively prime to a number 'n' or divided it.
(b) A product is divisible by a prime 'p' only when 'p' divides one of the factors.
(c) Every n > 1 is divisible by some prime.Theorem: Every compositely number could be factored into prime factors and each of these are unique in nature.

Determination of Prime factors:
The actual determination of the factorization of numbers into prime factors is of great importance to number theory.
When prime factor 'p' is found, then 'n' = pm and we can determine the factorization of the smaller number 'm'.
If a number is composite it must have a factor not exceeding $\sqrt{n}$.

We can find prime factorization by making a factor tree.
(a) Find any pair of factors.
(b) Find pairs of factors for the factors until we find all the factors as prime numbers.Example:

5 is a whole number as well as prime number. It is only divisible by itself and 1.

13 is a prime number as it is only divisible by itself and 1. It is also a whole number.

Factors:

Factors are the numbers which are multiplied to get another number. It is also known as Multiple.

Example:

20 = 4 x 5

In above equation , 4 and 5 are the factors of 20

50 = 25 x 2

Here , 25 and 2 are the factors of 50

What is Prime Factorization?

In number system, the prime factors of a positive integer are the prime numbers that divide that integer exactly, without leaving a remainder.
The process of finding these numbers is called integer factorization, or prime factorization.
The prime factorization of a positive integer is a list of the integer's prime factors, together with their multiplicity.

Prime Factorization Chart

Below you could see prime factorization chart

Prime factorization "is a method of finding the prime factors which is required to find the original number ".
In this prime factorization, the factors will always be the prime number.

Prime Factorization Definition

Prime factorization is the process of finding a list of prime factors for a number. To write in short each factor which is repeated can be written in exponent form. E.g.-

Prime factorization of a few numbers is shown below:

prime factorization of 24 = 2 x 2 x 2 x 3= 2³ x 3

prime factorization of 72 = 2 x 2 x 2 x 3 x 3 = 2³ x 3²

prime factorization of 98 = 2 x 7 x 7= 2 x 7²

prime factorization of 36 = 2 x 2 x 3 x 3 = 2² x 3³

prime factorization of 48 = 2 x 2 x 2 x 2 x 3 = 2⁴ x 3

prime factorization of 100 = 2 x 2 x 5 x 5 = 2² x 5⁵

Prime Factorization Tree

Factorial tree is another method of finding prime factorization for a number. In a factor tree, we start a number and write its two factors as branches. For example suppose we start with 30. 30 = 5 x 6. Then we take its factors and break them into branches if possible. Like, here we can break 6 into 2 and 3. We continue till we have prime numbers in the last nodes. Prime factorization is the product of numbers in the factorial trees.

Examples using factorial tree:

We could also make another factorial tree for same number. Like 30 = 2 x 15. Now we can further break 15 into 3 and 5.Final result is same in both cases.
Prime Factorization Tree

We can thus write the prime factorization of 30 as 2 x 3 x 5.

How to do Prime Factorization?

Below are the steps that explains how to do prime factorization-

Step 1:

Identify the given number is prime number or not.

Case 1:- If the given number is a prime number then the prime factor of this is only the same number.

Case 2:- If the given number is not a prime number, go after the following steps to find the prime factors of the given number.

Step 2:

Start with the first prime number 2.

Divide the given number by 2

Case 1:- if it has, no remainder then go to step 3.

Case 2:- if it has, a remainder then use next prime number to divide until you get zero as a remainder .

The prime divisor, which produces 0 as remainder is a prime factor.

Step 3:

Case 1:- If the answer in the step 2 is a prime number then it is also a prime factor.

Case 2:- If the answer in the step 2 is not a prime number start with step 2 and repeat the same procedure.

Prime Factorization using Exponents

The product of a number, when repeated more than once can be expressed in a simple way. It is known as exponential form. This form is very helpful in expressing large numbers. 6 × 6 is expressed in brief 6². In 6², 6 is called the base, 2 is called the index or the power or the exponent.62 is read as ‘6 squared’ or 6 raised to the power 2.

To write prime factorization of 147 using exponents?

147 ÷ 3 = 49

49 ÷ 7 = 7

All the factors are prime numbers write in exponents

147 = 3 × 7 × 7 = 3 × 7²