. Fundamental Theorem of Arithmetic Let us consider the following conversation between a Teacher and students. Teacher : Factorise the number . Malar : × Raghu : × Iniya : × Kumar : × Malar : Whose answer is correct Sir?
Now, count the number of ’s, ’s and ’s. Malar : I got four ’s, one and one . Raghu : I got four ’s, one and one . Iniya : I also got the same numbers too.
Kumar : Me too sir. Malar : All of us got four ’s, one and one . This is very surprising to us. Teacher : Yes, It should be.
Once any number is factorized up to a product of prime numbers, everyone should get the same collection of prime numbers. This concept leads us to the following important theorem. Theorem (Fundamental Theorem of Arithmetic) (without proof) “Every positive integer (except the number ) can be represented in exactly one way apart from rearrangement as a product of one or more primes.” The fundamental theorem asserts that every composite number can be decomposed as a product of prime numbers and that the decomposition is unique. In the sense that there is one and only way to express the decomposition as product of primes.
In general, we conclude that given a composite number N, we decompose it uniquely in the form N q n where p p p p n ,..., are primes and q q q q n ,..., are natural numbers. First, we try to factorize N into its factors. If all the factors are themselves primes then we can stop. Otherwise, we try to further split the factors which are not prime.
Continue the process till we get only prime numbers. Illustration For example, if we try to factorize 32760 we get 32760 = × × × × × × × Thus, in whatever way we try to factorize 32760 , we should finally get three ’s, two ’s, one , one and one . The fact that “Every composite number can be written uniquely as the product of power of primes” is called Fundamental Theorem of Arithmetic . .
. Significance of the Fundamental Theorem of Arithmetic The fundamental theorem about natural numbers except , that we have stated above has several applications, both in Mathematics and in other fields. The theorem is vastly important in Mathematics, since it highlights the fact that prime numbers are the ‘Building Blocks’ for all the positive integers. Thus, prime numbers can be compared to atoms making up a molecule.