Question

Prove that positive integer n is prime if number prime p is less than or equal to squre root of divides n

Answer 1

Let n be a composite number ( Numbers those have factors other than 1 and no. itself ). Let the factors be p and q.

So, n = pq where we assume q < p

According to question, p ≤ √n

Therefore, q < p ≤ √n

Hence q < √n ___(1)

If p is smaller than √n i.e., p < √n __(2)

Then (1) × (2)

pq < √n • √n

n < n

This leads to a contradiction because a number can't be smaller than itself.

Hence, n is a prime number where no.other prime number p less or equal to √n can divide n.

Q.E.D