Let any integer n ≥ 0 is a composite number.
so, n has a factor between 1 to n.

Let r is the factor of n, such that, 1 < r="" />< />

so, we can write n = rs , where r and s are positive integers such that, 1 < r,="" s="" />< />

assume , integer r is greater than equal to s.
e.g., r ≤ s

And also consider s > √n

so, √n < s="" ≤="" />

it means, r > √n

but n = rs > √n. √n

so, n > n which is a contradiction.
hence, our assumption was wrong.
therefore, a positive integer n is prime number,if no prime p less than or equal to √n divides n

**Proof by Contradiction:**

We will prove the given statement by contradiction.

Assume that there exists a positive integer n that is not a prime number, even though no prime p less than or equal to the square root of n divides n.

**1. Definition of a Prime Number:**

A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself.

**2. Assumption:**

Let's assume that n is not a prime number, meaning it has a positive divisor other than 1 and itself.

**3. Divisors of n:**

Since n is not a prime number, it has at least one positive divisor other than 1 and itself. Let's call this divisor d, where d is greater than 1 and less than n.

**4. Divisor d:**

Since d is a divisor of n, it means that d divides n without leaving any remainder. Therefore, we can write n as a product of d and another positive integer k, where n = d * k.

**5. Proof by Contradiction:**

Now, our assumption is that no prime p less than or equal to the square root of n divides n. However, since d is a divisor of n, it means that d is less than or equal to n. Therefore, d must be less than or equal to the square root of n.

**6. Prime Divisor:**

Since d is less than or equal to the square root of n, it contradicts our assumption that no prime p less than or equal to the square root of n divides n. This implies that d itself must be a prime number.

**7. Conclusion:**

From step 6, we have found a prime divisor (d) of n, which contradicts our assumption that n has no prime divisors less than or equal to the square root of n. Therefore, our assumption that n is not a prime number must be false.

**8. Final Result:**

Hence, we can conclude that if no prime p less than or equal to the square root of n divides n, then n must be a prime number.