Question

explain uniqueness of factorisation theorem using an example

Answer 1

Unique prime factorisation theorem

Interesting topic . So let us start ..

Firstly let us talk about prime numbers .

Primes :

Prime numbers are numbers which has exactly 2 factors .The 2 factors are the number and itself . Take an example - say 5 . 5 has 2 factors : 5 and 1 and hence the 2 factors are the number 5 itself and 1 .

Now let us get into the prime factorization theorem :

The prime factorization theorem states that any natural number N ( such that N is more than 1 ) can be expressed as the product of prime numbers with powers .

O_o Let us make that simple :

Take an example again so that our work is easier .

Example

Let us say that the given number is 8 .

So according to the theorem above , we can express 8 as the power of product of primes .

8 = 2 × 2 × 2

⇒ 8 = 2³ and hence the theorem is true !

Another example : Take the number 18 .

18 = 9 × 2

⇒ 18 = 3² × 2 and hence the theorem is explained .

This theorem can be applied to many problems . I am solving one so as to clear the concept .

QUESTION :

What are the primes such that the sum of the primes is equal to the cube of the difference between them ?

ANSWER :

Let the primes be a and b .

According to the question we have :

a + b = ( a - b )³ .. Note that a > b

Let a - b be something say x .

a + b will be hence x + 2 b .

Then we substitute the value and we get :

x + 2 b = x³

⇒ x³ - x = 2 b

Take commons :

⇒ x ( x² - 1 ) = 2 q

⇒ x ( x + 1 )( x - 1 ) = 2 q

By uniqueness of prime factorisation we will have :

⇒ x ( x + 1 )( x - 1 ) = 1 × 2 × q

So basically we can compare the RHS and LHS to get :

x - 1 = 1

x + 1 = q

x = 2

x - 1 = 1 ⇒ x = 1 + 1 = 2 and hence the two relations are same .

x = q - 1 ⇒ 2 = q - 1 ⇒ q = 2 + 1 = 3

The value of q is 3 .

p + q = x³

⇒ p + q = 2³ = 8

⇒ p + 3 = 8

⇒ p = 8 - 3 ⇒ p = 5

The two primes are 5 and 3 .

Answer 2

ANSWER

Uniqueness of factorisation theorem.

Step By Step Explanation

First out all we will know the meaning of factorisation of any term.

Factorisation :> Factorisation or factoring consists of number or another mathematical object which are usually the product of several smaller or simpler factors of same kind.

-The numbers which we get on factorisation of an term gives the factors of that main term.

-And when the factors are presented in such a form in which the numbers are prime only is called

Prime Factorisation.

-Prime Numbers

Prime Numbers are those real numbers which has a factor one and number itself only.

Example > 2 , 3 , 5 , 7 ......

-Proof for prime Numbers

3 = $3 \times 1$

5 = $5 \times 1$

We can see here that prime numbers can only be factorised into one and number itself.

Now we shall return to our question, UNIQUENESS OF FACTORISATION THEOREM :>

1)It states that any natural number greater than one has a prime factorisation.

2)And prime factorisation of any natural number greater than one is unique.

Now,we have the prime factorisation of any natural number is unique except its order.

Now let a composite number be (C)

And, C can be factorised in form of primes such that =>

C =$p1\times p2 \times p3$.....pn

where p1 , p2 , p3 and pn are prime numbers and are ascending order i.e p1 is equal to or less than to p2 .... is similar to or less than pn.

-If we combine the Same primes we get power of primes.

-And the numbers are arranged in an ascending order to get an unique factorisation.

EXAMPLES

1)12 = $2 \times 2 \times 3= 2^2 \times 3$

2)20 = $2 \times 2 \times 5 = 2^2 \times 5$

3)15 = $3\times 5$

4)9 = $3\times 3 = 3^2$

This all shows that the factorisation of any natural number more the 1 is unique.