Question

Prove that a positive integer $n$ is prime number, if no prime $p$ less than or equal to $\sqrt{n}$ divides $n.$

Answer 1

Let $n$ be a positive integer such that no prime less then to $\sqrt{n}$ 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 $1<a\le b$
$\Rightarrow a\le \sqrt{n}$ and $b\ge \sqrt{n}$
Let $p$ be a prime factor of $a$. Then $p\le a\le \sqrt{n}$ and $p|a$
$\Rightarrow$ $p|ab$
$\Rightarrow$ $p|n$
$\Rightarrow$ a prime less than $\sqrt{n}$ divides $n$
This contradicts our assumption that no prime less than $\sqrt{n}$ divides $n$. So, our assumption is wrong. Hence, $n$ is a prime.