Real Numbers-The Fundamental Theorem of Arithmetic


We have already studied in the previous classes that any natural can be written as a product of its prime factors. e.g. = 3 = 3, 6 = 2 \displaystyle \times 3, 275 = 11 \displaystyle \times 25 etc. i.e, any natural number can be obtained by multiplying prime numbers.

If we take prime numbers like 2,3,5,7,11,13, and 29 and if we multiply some or all of these numbers, allowing them to repeat as we can, then we can get a large collection of positive integers (infinitely).

 e.g.             11 \displaystyle \times 13 \displaystyle \times 29 = 4147

 7 \displaystyle \times 11 \displaystyle \times 13 \displaystyle \times 29 = 29029

 5 \displaystyle \times 7 \displaystyle \times 11 \displaystyle \times 13 \displaystyle \times 29 = 145145

 3 \displaystyle \times 5 \displaystyle \times 7 \displaystyle \times 11 \displaystyle \times 13 \displaystyle \times 29 = 435435

 2 \displaystyle \times 3 \displaystyle \times 5 \displaystyle \times 7 \displaystyle \times 11 \displaystyle \times 13 \displaystyle \times 29 = 870870

 22 \displaystyle \times 3 \displaystyle \times 5 \displaystyle \times 7 \displaystyle \times 11 \displaystyle \times 13 \displaystyle \times 29 = 1741740

 23 \displaystyle \times 3 \displaystyle \times 5 \displaystyle \times 7 \displaystyle \times 11 \displaystyle \times 13 \displaystyle \times 29 = 3483480

Now, we see there are infinitely many primes and so we can get an infinite collection of numbers by all the primes and all possible products of primes.

In the other hand, if we factorise positive integers, we have to do the opposite of what we have done so far.

We can use factor tree for factorise of 65520 as shown below.

Hence, we have factorised 65520 as 2 \displaystyle \times 2 \displaystyle \times 2 \displaystyle \times 2 \displaystyle \times 3 \displaystyle \times 3 \displaystyle \times 5 \displaystyle \times 7 \displaystyle \times 13 as a product of primes, i.e., 65520 = 24 \displaystyle \times 32 \displaystyle \times 5 \displaystyle \times 7 \displaystyle \times 13 as a product of powers of primes.

If we take 123456789 then it is equal to 32 \displaystyle \times 3803 \displaystyle \times 3607.

In these factors 3803 and 3607 are primes. It means we have to understand that every composite number can be written as the product of primes. It is called Fundamental Theorem of Arithmetic because of its basic crucial importance to the study of integers.

Hence, we can say Fundamental Theorem of Arithmetic as follows :

Every composite number can be expressed (factorised) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur, i.e., for any composite number there will be one and only one way to express it as a product of primes, as long as we are not particular about the order in which the primes occur.

e.g    2 \displaystyle \times 3 \displaystyle \times 5 \displaystyle \times 7 \displaystyle \times 11 as the same as 5 \displaystyle \times 7 \displaystyle \times 11 \displaystyle \times 2 \displaystyle \times 3 or any other possible order in which these primes are written. It can be defined as follows :

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

If any composite number is x, then the factor of x = p1 , p2 , p3 , …. Pn, where p1, p2, p3, …. Pn are primes and written in ascending order, i.e., \displaystyle {{p}_{1}}\le {{p}_{2}}\le {{p}_{3}}......\le {{p}_{n}}.

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

E.g.    65520 = 2 \displaystyle \times 2 \displaystyle \times 2 \displaystyle \times 2 \displaystyle \times 3 \displaystyle \times 3 \displaystyle \times 5 \displaystyle \times 7 \displaystyle \times 13

24 \displaystyle \times 32 \displaystyle \times 5 \displaystyle \times 7 \displaystyle \times 13

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

There are many applications of the Fundamental Theorem of Arithmetic in mathematics as well as in other fields.

Example 4:Consider the number 16n, where n is a natural number. Check whether there is any value of n for which 16n ends with the digit zero.

Solution. If the number 16n, for any n, were to end with the digit zero, then it would be divisible by 5 and so its prime factorization must contain the prime 5.

16n = (24)n = 24n

The only prime in the factorization of 16n is 2.

There is no other primes in the factorization of 16n = 24n

[By uniqueness of the Fundamental Theorem of Arithmetic]

5 does not occur in the prime factorization of 16n for any n.

4n does not end with the digit zero for any natural number n.

We have already learnt how to find the HCF and LCM of two positive integers using the Fundamental Theorem of Arithmetic in previous classes, without realising it. This method is also called the Prime factorization method.

Example :

Find the LCM and HCF of 18 and 60 by the prime factorization method.


18 = 2 \displaystyle \times 3 \displaystyle \times 3 = 21 \displaystyle \times 32

60 = 2 \displaystyle \times 2 \displaystyle \times 3 \displaystyle \times 5 = 22 \displaystyle \times 31 \displaystyle \times 51

We get HCF of 18 and 60 as 2

And LCM of 18 and 60 as 180

But now, we have to understand that

HCF of 18 and 60 is 21 = Product of the smallest power of each common prime factor in the numbers.

LCM of 18 and 60 is 22 \displaystyle \times 32 \displaystyle \times 5 = Product of the greatest power of each Prime factor, involved in the numbers.

For any two positive integers a and b , (HCF of a and b) \displaystyle \times (LCM of a and b) = a x b

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