Real numbers The Fundamental Theorem of Airthmetic

The Fundamental Theorem of Airthmetic

In your earlier classes, you have seen that any natural number can be written as a product of its prime factors. For instance , 2 = 2, 4 =2 x2 , 253, and so on. Now,let us try and look at natural numbers from the other direction. That is, can any natural number be obtained by multiplying prime numbers ? Let us see.

Take any collection of prime numbers , say 2,3,7,11 and 23. If we multiply some or all of these numbers, allowing them to repeat as many times as we wish. We can produce a large collection of positive integers (In fact , infinitely many). Let us list a few.
7x11x23 = 1771 3x7x11x23 = 5313
2x3x7x11x23 = 10626 23 x 3x73= 8232
22 x3x7x11x23 = 21252

And so on.

Now, let us suppose your collection of primes includes all the possible primes. What is your guess about the size of this collection? Does it contain only a finite number of integers, or infinitely many ? Infact, there are infinitely many primes. So, if we combine all these primes in all possible products of primes. The question is – can we produce all the composite numbers this way ? What do you think ? Do you think that there may be a composite number which is not the product of powers of primes ? Before we answer this, let us factorise positive integers, that is, do the opposite of what we have done so far.

We are going to use the factor tree with which you are all familiar. Let us take some large number , say 32760 and factorise it as shown :

So we have factorised 32760 as 2 x2x 2x3x3x5x7 x13 as a product of primes ie, 32760 = 23 x32 x5 x 7 x 13 as a product of powers of primes. Let us try another number , say , 123456789. This can be written as 32 x 3803 x 3607 . Of course, you have to check that 3803 and 3607 are primes. (Try it out for several other natural numbers yourself.) This leads us to a conjecture that every composite number can be written as the product of powers of primes. In fact, this statement is true, and is called the Fundamental Theorem of Airthemetic because of its basic crucial importance to the study of integers. Let us now formally state this theorem.