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Fundamental Theorem of Arithmetic

We know that the set of Natural numbers can be broadly classified into Prime Numbers and Composite Numbers. Prime numbers is a number which has no factors other than 1 and itself. Any natural number which has is not a prime number i.e. which has at least one factor other than 1 and itself is called a composite number.

Now, the fundamental theorem of arithmetic says that, "Every composite number can be expressed as a product of prime factors in a unique way."

What this means is that, any composite number could be written as a product of some prime numbers and this combination of prime numbers for that composite number is unique i.e. no other combination of prime numbers is possible whose product can give that composite number.

For example, 12 could be written as a product of prime numbers as 2x2x3. No other combination of prime numbers is possible which can  give 12 as a product. Note that every composite number hence has one and only one such combination of prime numbers whose product gives the composite number. 

Simple it might look, the outcome of this theorem is fabulous. Every composite number being a product of a unique combination of prime numbers implies that prime numbers are the generators of composite numbers! In other words, prime numbers are the basic building blocks of natural numbers. Prime numbers are for natural numbers what vowels are for alphabets.

The set of natural numbers could be expressed only in terms of prime numbers as follows: 
N = {1,2,3,2x2,5,2x3,7,2x2x2,3x3,2x5,11,2x2x3,13,....}

-by Gurudev
MADE IN INDIA
gurudevp@vsnl.net
On 19 May 2003

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