The Fundamental Theorem of Arithmetic

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 × 2, 253 = 11 × 23, 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:

7 × 11 × 23 = 1771 3 × 7 × 11 × 23 = 5313

2 × 3 × 7 × 11 × 23 = 10626 23 × 3 × 73 = 8232

22 × 3 × 7 × 11 × 23 = 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 ways, we will get an infinite collection of

numbers, all the primes and 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 × 2 × 2 × 3 × 3 × 5 × 7 × 13 as a product of

primes, i.e., 32760 = 23 × 32 × 5 × 7 × 13 as a product of powers of primes. Let us try

another number, say, 123456789. This can be written as 32 × 3803 × 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 Arithmetic because of its basic crucial importance to the study of integers. Let us now formally state this theorem.

Theorem 1.2 (Fundamental Theorem of Arithmetic): Every composite number

can be expressed (factorised) as a product of primes, and this factorisation is

unique, apart from the order in which the prime factors occur. The Fundamental Theorem of Arithmetic says that every composite number

can be factorised as a product of primes. Actually it says more. It says that given

any composite number it can be factorised as a product of prime numbers in a

‘unique’ way, except for the order in which the primes occur. That is, given any

composite number there is one and only one way to write it as a product of primes,

as long as we are not particular about the order in which the primes occur. So, for

example, we regard 2 × 3 × 5 × 7 as the same as 3 × 5 × 7 × 2, or any other

possible order in which these primes are written. This fact is also stated in the

following form:

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

of its factors.

In general, given a composite number x, we factorise it as x = p1 p 2 ... pn, where

p1

, p2

,..., pn are primes and written in ascending order, i.e., p1

£ p2

£ . . . £ p

n

. If we combine the same primes, we will get powers of primes. For example,

32760 = 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13 = 23 × 32 × 5 × 7 × 13

Once we have decided that the order will be ascending, then the way the number

is factorised, is unique.

The Fundamental Theorem of Arithmetic has many applications, both within

mathematics and in other fields. Let us look at some examples.

Example 5: Consider the numbers 4n, where n is a natural number. Check whether

there is any value of n for which 4n ends with the digit zero.

Solution: If the number 4n, for any n, were to end with the digit zero, then it would be divisible by 5. That is, the prime factorisation of 4n would contain the prime 5.

not possible because 4n = (2)2n; so the only prime in the factorisation of 4n is 2. So, the

uniqueness of the Fundamental Theorem of Arithmetic guarantees that there are no

other primes in the factorisation of 4n. So, there is no natural number n for which 4n

ends with the digit zero.

You have already learnt how to find the HCF and LCM of two positive integers

using the Fundamental Theorem of Arithmetic in earlier classes, without realising it!

This method is also called the prime factorisation method. Let us recall this method

through an example.

Example 6: Find the LCM and HCF of 6 and 20 by the prime factorisation method.

Solution: We have: 6 = 21 × 31 and 20 = 2 × 2 × 5 = 22 × 51.

You can find HCF (6, 20) = 2 and LCM (6, 20) = 2 × 2 × 3 × 5 = 60, as done in your

earlier classes.

Note that HCF (6, 20) = 21 = Product of the smallest power of each common

prime factor in the numbers.

LCM (6, 20) = 22 × 31 × 51 = Product of the greatest power of each prime factor, involved in the numbers.

From the example above, you might have noticed that HCF (6, 20) × LCM (6, 20)

= 6 × 20. In fact, we can verify that for any two positive integers a and b,

HCF (a, b) × LCM (a, b) = a × b. We can use this result to find the LCM of two

positive integers, if we have already found the HCF of the two positive integers.

Example 7: Find the HCF of 96 and 404 by the prime factorisation method. Hence,

find their LCM.

Solution: The prime factorisation of 96 and 404 gives:

96 = 25 × 3, 404 = 22 × 101

Therefore, the HCF of these two integers is 22 = 4.

Also, LCM (96, 404) =

96 404 96 404

9696

HCF (96, 404) 4

Example 8: Find the HCF and LCM of 6, 72 and 120, using the prime factorisation

method.

Solution: We have:

6 = 2 × 3, 72 = 23 × 32, 120 = 23 × 3 × 5

Here, 21 and 31 are the smallest powers of the common factors 2 and 3, respectively.

So, HCF (6, 72, 120) = 21 × 31 = 2 × 3 = 6

23, 32 and 51 are the greatest powers of the prime factors 2, 3 and 5 respectively

involved in the three numbers.

So, LCM (6, 72, 120) = 23 × 32 × 51 = 360

Remark: Notice, 6 × 72 × 120 ¹ HCF (6, 72, 120) × LCM (6, 72, 120). So, the

product of three numbers is not equal to the product of their HCF and LCM.