# The Fundamental Theorem of Arithmetic

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

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**Example :**

756 = 2^{2}Ã— 3^{3}Ã— 7

3003 = 11 Ã— 7 Ã— 13 Ã— 3

6460 = 2 Ã— 5 Ã— 17 Ã— 19 Ã— 2

1218 = 2 Ã— 3 Ã— 7 Ã— 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.

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so we can factorize 64890 = 2 Ã— 5 Ã— 3 Ã— 3 Ã— 7 Ã— 103

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = 2 Ã— 3^{2}Ã— 5 Ã— 7 Ã— 103

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

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This is the Fundamental Theorem of Arithmetic.

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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 = p_{1} p_{2 â€¦}p_{n} where p_{1}, p_{2},â€¦, p_{n} are prime numbers in the ascending order (i.e.) p_{1}â‰¤ p_{2}â€¦â‰¤ p_{n.} 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.