Prove that a positive integer $n \geq 2$ is prime if and only if there is no positive integer greater than $1$ and less than or equal to $\sqrt{n}$ that divides $n$

Proof given by contradiction.

Step-by-step explanation:

Given that:

To prove:

is prime if and only if no positive integer > 1 and divides .

Solution:

First of all, let is a composite number i.e. not a prime number such that:

and and are prime and divides and also divides .

Let

or

1. :

is prime and is a divisor of .

2. :

We have assumed that

is a prime number and is a divisor of .

But we are given that no prime number divides but we have proved that divides .

So, it is a contradiction to our assumption.

Therefore, our assumption is wrong that   is a composite number.

Hence, proved that is a prime number.

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step-by-step explanation: