Factorization

In the case of integers, ancient Greek mathematicians were the first to examine factorization. They proved the arithmetic fundamental theorem, which claims that any positive integer can be factored into a product of prime numbers that can’t be factored into integers larger than one. 

Prime factorization can be used to express a number as a product of its prime factors. A prime number has only one and the number itself as components. Suppose we have the number 20 as an example and we will find the prime factorization of the number. We know that 20 = 5 × 4 is a prime number, but 4 is not a prime number. The number 4 can be factored further as 2 × 2 with 2  being a prime number. As a result, the prime factorization of 20 is 2 × 2 × 5  with all factors being prime numbers.

WHAT IS PRIME FACTORISATION:-

A process of representing the given number as the product of the prime numbers is known as prime factorization of the given number. The numbers containing the one element, one, and the number itself are known as the prime numbers. Some of the prime numbers are 5, 7, 11, 13, 17……, and many more numbers. The representation of a number as a product of prime numbers is called “prime factorization.” The prime factorization of 50, for example, can be done as follows:

We know that 50 = 10 × 5 is a prime number, but 10 is not a prime number. The number 10 can be factored further as 2 × 5 with 2 and 5 being prime numbers. As a result, the prime factorization of 50 is 2 × 5 × 5  with all factors being prime numbers.

Prime factorization is the process of breaking down a number into prime numbers that help construct the number when multiplied. To put it another way, prime numbers are multiplied to get the original number.

METHODS OF PRIME FACTORIZATIONS:-

The prime factorization of a number can be done in a variety of ways. The most commonly used prime factorization methods are listed below:-

1. Division method of prime factorization
2. Factor tree method of prime factorization

We will discuss both methods one by one:-

Division method of prime factorization

 The steps below are used to determine a number’s prime factorization using the division method:-

Step 1:- Divide the number by the least prime number in such a way that the smallest prime number entirely divides the number.

Step 2:- Divide Step 1’s quotient by the smallest prime number once again.

Step 3:- Repeat the same as step 2 until the quotient becomes 1.

Step 4:- Lastly, multiply all of the divisors’ prime factors.

Factor tree method of prime factorization

The factor tree approach involves finding a number’s factors and then factorizing those numbers until we reach prime numbers. The steps below are used to determine a number’s prime factorization using the factor tree method:

Step 1:- Put the number on the top of the tree.

Step 2:- Then, as the tree’s branches, write down the relevant pair of factors.

Step 3:- Factorize the obtained composite factors discovered in step 2 and record the pair of factors like the tree’s next branches.

Step 4:- Repeat the same as step 3 until all of the composite factors’ prime factors are found.

APPLICATION OF PRIME FACTORISATION:-

In the actual world, prime factorization is commonly employed. The following are the two most common uses for prime factorization:-

1.  :- Cryptography is a code-based way of securing data. Prime factorization is useful for coders who want to make a unique code out of numbers that aren’t too large for computers to store or process rapidly.
2. To find the HCF and LCM of the given number:- The prime factorization method is used to identify the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of two numbers. To accomplish so, we must first factorize both integers into prime factors. It’s important to remember the following points about HCF and LCM:

• HCF is the product of each common prime factor’s smallest power.
• LCM is the product of each common prime factor’s highest power.

CONCLUSION:- 

The process of factoring the given number in terms of the prime numbers, with prime numbers as the factors, is known as the prime factorization of the given number. Prime factorization is the method of determining the given number’s prime factors so that the original number is often divisible by these factors. 

 To find the prime factors of an integer, the easiest approach is to divide the original number by prime factors until the remainder equals 1.