Question

Explain the fundamental theorem of arthamatics

Answer 1

Hello dea!!

Here is ur answer!!

》FUNDAMENTAL THEOREM OF ARTHAMETICS

Every composite number can be expressed as product of primes and this factorization is unique a part from the order in which the prime no occurs.

Example : 18=2×3×3

》THEOREM

Let P be a prime no.

If P divides a² then

p/a,

where P is a +ve integer.

》GIVEN : P/a² , a : +ve integer.

》TO PROVE : P/a

》PROOF : a = P¹ × P² × P³........Pn

a² = ( P¹ × P² × P³.....Pn ) (P¹ × P² × P³.....Pn)

= ( P¹ × P² × P³ ......Pn)²

But P is one of the prime factor of a² ( By F.T.A)

Prime factors of a² = P¹ , P² ,P³.....Pn ( By U.F.T.A)

Therefore, P is one of P¹ , P² , P³......Pn
》P divides a. (P/a)

Hope it helps..!!

Answer 2

Hey friend !

Before going to the main concept , let us see what are prime numbers.

What are prime numbers actually ?
Answer :-
Prime numbers are those numbers which have only 2 factors - 1 and the number itself.
Example : 2 , 3 , 5 , 7 , 11 etc .. are prime numbers.
they have only 2 factors.

Natural numbers :- 1 , 2 , 3 , 4 etc ...

Now lets see what are composite numbers ..

Composite no:s => numbers that have more than 2 factors.


WE have learnt in our previous classes , that natural numbers can be written as the product of its prime factors .
For example , lets take a natural number , say 253.
IT can be expressed as ,
253 = 11 x 23

This factorization is unique , which means that no other prime numbers can take place in the place of 11 and 23.


Let us take an another example :
Lets take a number say , 32760.

Prime factorization :-
32760 = 2 x 2 x 2 x 3 x 3 x 5 x 7
or
32760 = 2³ × 3 ² × 5 × 7


We get to know that every composite number can be expressed as the product of its prime numbers and this factorization is unique , but they can be expressed in any order.


So lets see what the fundamental theorem of arithmetic states :-
"Every composite number can be expressed [ factorized ] as a product of its primes , this factorization is unique , apart from the order in which the prime factors occurs."

We can express a composite number as a product of prime factors ..... they can be in any order ... but the numbers that are the prime factors of it cannot be changed ..

Hope this Helps you !