Question

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

Answer 1

Let us prove the result using contradiction
Let $n\ge 0$ be a composite number
$\therefore$ n has a factor a such that $1<a<n$
We can write $n=ab$ where $a$ and $b$ are positive integers and $1<a,b<n$
We may assume that $b\le a$
let $b>\sqrt{n}$ .....1
Then $\sqrt{n}<b\le a\Rightarrow \sqrt{n}<a$
i.e $a>\sqrt{n}$....2
$\therefore n=ab>\sqrt{n}\times \sqrt{n}=n$
$\Rightarrow n>n$ which is contradiction
Hence our supposition was wrong.
Thus, for every positive integer n prime, if no prime p less than or equal to root n divides n.