Answer

Verified

414.6k+ views

**Hint:**In certain questions of proving statements we use either the direct step by step proof or assuming its contradiction or contrapositive method.

Here we use the contradiction method by assuming the opposite of the given statement and proving it is wrong. Thus as we need to prove the property of a prime number n, we assume n to be a composite number with the same given property and prove our assumption is wrong.

Prime number n indicates that the only factors of n are 1 and n itself. The numbers except primes with factors other than 1 and n are called composite numbers. Elaborating the previous sentence we can say a composite number n will have a factor x in between 1 and n such that n=xy where x and y are positive integers and factors of n.

**Complete step-by-step answer:**

Step 1: Let us assume the contradiction of the statement which is let $n \geqslant 0$ be a composite number. Thus n has a factor x such that $1 < x < n$ .

So we can write n=xy where x and y are positive integers such that $1 < x,y < n$

Let us assume $y \leqslant x$ (can be assumed either way)

As we are assuming the contradiction, let $y > \sqrt n $ … (1)

Thus $\sqrt n < y < x$ implies $\sqrt n < x$ or $x > \sqrt n $ … (2)

From formula (1) and formula (2) , $n = xy > \sqrt n \times \sqrt n = n$

Implies n>n which is a contradiction.

Hence our assumption is wrong.

Thus a positive integer n is prime, if no prime p less than or equal to $\sqrt n $ divides n.

Hence proved.

**Note:**Students may get confused with the inequality signs, they should keep in mind that

< denotes less than and

> denotes greater than.

Examples of prime numbers: 2,3,5,7,11,13,17,19,23,29,31,33, and so on.

Recently Updated Pages

How do you find slope point slope slope intercept standard class 12 maths CBSE

How do you find B1 We know that B2B+2I3 class 12 maths CBSE

How do you integrate int dfracxsqrt x2 + 9 dx class 12 maths CBSE

How do you integrate int left dfracx2 1x + 1 right class 12 maths CBSE

How do you find the critical points of yx2sin x on class 12 maths CBSE

How do you find the general solution to dfracdydx class 12 maths CBSE

Trending doubts

Give 10 examples for herbs , shrubs , climbers , creepers

Difference Between Plant Cell and Animal Cell

Write a letter to the principal requesting him to grant class 10 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Name 10 Living and Non living things class 9 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

List some examples of Rabi and Kharif crops class 8 biology CBSE

Write the 6 fundamental rights of India and explain in detail