Prove that a positive integer n …

According to the question,we have to prove that a positive integer n is prime number, if no prime p less than or equal to {tex}\sqrt{n}{/tex} divides n.

Let n be a positive integer such that no prime less than or equal to {tex}\sqrt {n}{/tex} divides n.
Then, we have to prove that n is prime. Suppose n is not a prime integer. Then, we may write n = ab where 1 < a {tex}\leq{/tex} b
{tex}\Rightarrow \quad a \leq \sqrt { n } \text { and } b \geq \sqrt { n }{/tex}
Let p be a prime factor of a. Then, {tex}p \leq a \leq \sqrt { n } {/tex} and p|a
{tex}\Rightarrow \quad p | a b{/tex}
{tex}\Rightarrow \quad p | n{/tex}
{tex}\Rightarrow{/tex} a prime less than {tex}\sqrt {n}{/tex} divides n.
This contradicts our assumption that no prime less than {tex}\sqrt {n}{/tex} divides n.
So, our assumption is is wrong. Hence, n is a prime.