Let x >= 2. Show that if x is not divisible by any positive integer n satisfying 2<= n<=sqrt(x) then x is a prime number.

Question

A proof is required, but I'm not sure how to do it. Could anyone guide me?

Answer 1

Hint

Suppose $x$ is composite. Then $x=ab$ for some $a, b$ such that $1< a,b<x$. If both $a,b$ were greater than $\sqrt{x}$, then $ab>x$, which is a contradiction. Thus every composite number has a divisor that is less than its square root.