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The Fundamental Theorem of Arithmetic

Any natural number can be written as a product of its prime factors.

 

Example :

756 = 22× 33× 7
3003 = 11 × 7 × 13 × 3

6460 = 2 × 5 × 17 × 19 × 2

1218 = 2 × 3 × 7 × 29 and so on.

 

If we take any collection of prime number allowing them to repeat as many times as we wish we get a large collection of positive integers.

But if a collection of prime numbers includes all possible prime numbers then we definitely get an infinite collection of numbers - all primes - and all possible products of primes.

Can there be a composite number which is not the product of powers of primes?

Using the factor tree, let us factorise some large number say 64890.
 


so we can factorize 64890 = 2 × 5 × 3 × 3 × 7 × 103

                                     = 2 × 32× 5 × 7 × 103

as a product of primes. This leads  us to conclude that every composite number can be written as a product of primes.

 

This is the Fundamental Theorem of Arithmetic.

 

Theorem 1.2

(Fundamental Theorem of Arithmetic) Every composite number can be expressed as a product of primes, and this factorisation is unique, apart from the order in which the factors occur.


Two theorems are stated in the above theorem namely:

1. Any composite number can be factorised as a product of primes. It further proceeds to say.

2. Any given composite number can be factorised as a product of prime numbers in a unique except in the order in which the primes occur.

The prime factorisation of a natural number is unique, except for the order of its factors.

Hence given a composite number x, we factorise it as x = p1 p2 …pn where p1, p2,…, pn are prime numbers in the ascending order (i.e.) p1≤ p2…≤ pn. If we combine the same prime we get the power of primes.

If we decide on that the order will be ascending then, the way the number is factorised is unique.





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