Fundamental theorem of Arithmetic

According to the fundamental theorem of arithmetic, “factorization of each and every composite number may be written as a product of primes regardless of the sequence in which the prime factors of that individual number occur.” The arithmetic basic theorem is a highly helpful way for understanding the prime factorization of any integer. “Every composite number can be factored as a product of primes, and this factorization is unique, aside from the order in which the prime components occur,” says the fundamental theorem of arithmetic. This theorem also states that this factorization must be one-of-a-kind. That is, there are no alternative ways to represent 240 as a prime product. We may, of course, adjust the order in which the prime factors appear. For example, the prime factorization may be written as: 240 = 31× 22 × 51 × 22 or 31 × 24 × 51 etc. However, the collection of prime factors (as well as the frequency with which each component appears) is unique.

Fundamental theorem of arithmetic formula:

To compute the HCF and LCM of two integers, the fundamental theorem of arithmetic is utilised. To begin, we must determine the prime factorisation of both integers. Following that, we’ll have a look at the following:

1. The lowest power of each common prime factor in two or more numbers is the HCF.
2. The greatest power of each common prime factor in the numbers is used to calculate the LCM of two or more integers.

Example: Using the prime factorisation method, calculate the HCF of 850 and 680.

We begin by determining the prime factorisations of these integers.

850=21×52×171

680=23×51×171

HCF is the product of each common prime factor’s lowest power.

HCF (850,680)=21×51×171=170

LCM is the greatest power of each common prime component multiplied by itself.

LCM(850,680)=23×52×171=3400

What is the fundamental theorem of arithmetic?

The statement of the basic arithmetic theorem is: Every natural number besides 1 can be factored as a product of primes, and this factorization is distinctive except for the sequence in which the prime factors are expressed. Every composite number could be distinctively decomposed as a product of prime numbers, according to the fundamental theorem. In the notion that the decomposition can only be stated in one way as a product of primes. In general, we discover that if we are given a composite number N, we can uniquely deconstruct it in this manner.  First, we attempt to factorise N into its constituents. If all of the components are primes, we can put it to an end. Otherwise, we try to split the non-prime elements further and repeat the procedure until we only have prime numbers.

Proof of fundamental theorem of arithmetic:

To show the fundamental theorem of arithmetic, we must prove the existence and uniqueness of prime factorisation. As a result, the fundamental theorem of mathematics asserts that proof requires two phases. We shall prove that the product of primes can only be represented in one way for any integer, n≥2.

n=a1.a2…ai

Step 1: Determine the existence of prime factorisation.

This will be demonstrated via mathematical induction. Mathematical induction is a technique for proving that a statement, formula, or theorem is true for all natural numbers.

The statement is accurate for n=2.

It demonstrates that if a statement is correct for the nth repetition (or number n), it is also true for the (n+1)th repetition (or number (n+1)).

Thus, the product of primes may be expressed as k. Let us now demonstrate that the assertion is accurate for n=k+1.

If k+1 is prime, the case is obvious.

Using the inductive step, j < k, k may be expressed as the product of primes. Because of (1), k+1 can alternatively be written as a prime product. The prevalence of factorisation is therefore demonstrated mathematically.

Step 2: The uniqueness of prime factorisation

Assume that n may be stated as the product of primes in two ways, for example,

n =a1a2…ai
    =m1m2…mj
m1m2…mj are coprime numbers meanwhile these are prime factorisations (as they are also prime numbers).

As a consequence, according to Euclid’s Lemma, a1 divides just one of the primes.

a1=m1 because m1 is the smallest prime.

Similarly, we may prove that an=mn  for all n. As a result, i=j

As an outcome, n′ prime factorisation is unique.