The Fundamental Theorem of Arithmetic is a mathematical theorem that states that every positive integer can be uniquely expressed as a product of prime numbers. In other words, we can say that every positive integer can be written as a unique product of primes, and that this representation is unique (up to the order of the factors).

## About The Fundamental Theorem of Arithmetic

This theorem is important because it gives us a way to break down any positive integer into its prime factors, which is useful for factorisation and many other mathematical and computational purposes. It is also an important result in number theory, which is the branch of mathematics that studies the properties of integers.

For example, the number 21 can be written as 3 x 7, and this is the unique prime factorization of 21. It cannot be written as a product of other primes, such as 2, 5, 11, etc.

### Video Representation of The Fundamental Theorem of Arithmetic

#### Application of The Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic has many important applications and implications in mathematics and other fields. It is also one of the important tools for class 10 Maths students. Some of the main uses of this theorem are:

**Prime factorization:**Prime factorisation is very common in all the classes. In class 10 the practice of factorisation speedup the calculation. As mentioned earlier, the Fundamental Theorem of Arithmetic allows us to express any positive integer as a unique product of prime numbers. This is useful for many purposes, such as finding the prime factorization of a number and very helpful for class 10 students. It can also be used to determine if a number is prime or composite, and to find the greatest common divisor (GCD) or Highest Common Factor (HCF) of two numbers as given in 10th Maths Exercise 1.1 and Exercise 1.2.**Cryptography:**In higher classes, the Fundamental Theorem of Arithmetic is also important in the field of cryptography, which is the study of techniques for secure communication. The main application of this theorem is the prime factorization or to break down large numbers into their prime factors, which can be used to encrypt and decrypt messages.

##### Use of Fundamental Theorem of Arithmetic in Number Theory

Number theory: Class 10 Maths, we study about the basics of the Fundamental Theorem of Arithmetic, which is an important result in number theory for higher education. The branch of mathematics that studies the properties of integers also consider it a helping hand. It is used to prove many other theorems and results in number theory, and it has also led to the development of many important concepts and techniques in this field.

###### Use of Fundamental Theorem of Arithmetic in Computational purposes

In class 10 Maths Exercises 1.2, the Fundamental Theorem of Arithmetic is used to calculate the LCM and HCF of two number. It is also used for many other computational purposes, such as finding the prime factorization of a number in order to determine if it is prime or composite numbers. It helps to find the GCD of two numbers. The Fundamental Theorem of Arithmetic is also useful for many applications, such as computer algorithms and software that perform mathematical calculations.

Overall, the Fundamental Theorem of Arithmetic is an important result in Maths for 10th Standard that has many applications and implications in various fields. It is a fundamental result that is widely used and has played a key role in the development of many important concepts and techniques during the factorizing the composite numbers.