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Fundamental Theorem of Arithmetic

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The fundamental theorem of arithmetic is one of the most important theorem which comes under number system. There are many different types of numbers like, whole number, integers number, real number, natural number, co-prime number and so on. It is also called as unique prime factorization. In this page we are going to learn about the FTA's statement, proof and applications.

Fundamental Theorem of Arithmetic Definition

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The definition of FTA is about the prime numbers which is also known as statement of FTA. Fundamental theorem of arithmetic Statement is given below,
Fundamental Theorem of Arithmetic states that every natural number greater than 1 can be written as a unique product of prime numbers.

Fundamental Theorem of Arithmetic Proof

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Theorem of the FTA is as follows:
Statement: A natural number would be either prime or is able to be factorized in the form of a unique product of prime numbers.
Proof: The FTA proof has divided in to two parts, 
Part 1: (Proof of Existence)

Here we use Euclid's Lemma

"if any prime number p divides two natural numbers x and y, then either p divides x or y or both".

From the above lemma prove that each integer > 1 is expressed as a product of prime numbers.

Let it is true for 2 to n.

From lemma (If n is a prime number, then represented in the form of product of one factor itself)

Hence proof (for a prime number)

But if n $
\neq $ prime, i.e. n = composite,

Again let consider two integers x and y,

n = x y

If $x = x_{1}.x_{2}....x_{m}$ and $y = y_{1}.y_{2}....y_{n}$ where 1 < x ≤ y < n,

x and y can be written as product o prime numbers ( From the above hypothesis of induction,)

$n = x y = x_{1}.x_{2}....x_{m} \times y_{1}.y_{2}....y_{n}$ 
Part 2: (Proof of Uniqueness)

Here we show that product of these prime numbers is unique.

Suppose that a number p (> 1) .......( by contradiction.)

This can be represented in the two different ways,

$p= x_{1}.x_{2}....x_{m} = y_{1}.y_{2}....y_{n}$

To prove uniqueness, one has to show that $x_{i}$'s are rearrangement of $y_{j}$'s, i.e. m = n.

$x_{1}$ divides one of the $y_{j}$; i.e. $x_{1}$ divides $y_{1}$. .....(by Euclid's Lemma,)

But we know that $y_{1}$ is a prime number, so the only divisors of $y_{1}$ are 1 and itself. Thus $x_{1}$ = $y_{1}$. Then,

$\frac{p}{x_{1}}$ = $x_{2}...x_{m}=y_{2}...y_{n}$

Similarly we have $x_{2}=y_{2}$ and we can write that

$\frac{p}{x_{1}.x_{2}}$ = $x_{3}...x_{m}=y_{2}...y_{n}$

This is applied on each of the x's and y's ,

Hence,

each $x_{i}=y_{j}$

Therefore, the uniqueness of fundamental theorem is proved.

Fundamental Theorem of Arithmetic Applications

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The applications of FTA has wide rand in current world but in mathematics it has a wide applications.

1) The FTA helps to understand the necessity and importance of primes in mathematics. According to FTA all the integers are composed of primes in some unique way. 

2) The prime factorization technique is the most important application of FTA which used in canonical representation of numbers. With the help of canonical representation many arithmetical functions are define. 

3) FTA is the central part of the number system. FTA is used to proof many properties and theorems of number system.

4) FTA is used in greatest common divisor and lowest common multiple.    

Many more other importance application of FTA are used in mathematics and science. 

Models Fundamental Theorem of Arithmetic

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The basic multiplicative models of FTA are as follows,

Lemma 1: There exists a 1-1 effectively calculable function v from the natural numbers onto mosaics.

Lemma 2: Let N be the set of all natural numbers i.e. strictly positive integers put $\psi^{2} = \psi(\psi(.))$ and define $\psi^{k}$,k >=3, recursively. then ${\psi^{i}:i\epsilon N}$ is an infinite recursive set of effectively calculable number theoretic function, all intimately related to FTA. 

Fundamental Theorem of Arithmetic Examples

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Some of the solved examples of FTA are given below,

Question 1:

The number 132 is to be written as a product of its prime factors. Which one of the following is correct?

Given: 132

Fundamental Theorem of Arithmetic Examples
Question 2:

The number 240 is to be written as a product of its prime factors. Which one of the following is correct?

Given: 240 

Fundamental Theorem of Arithmetic Problems