let n be a positive integer if n has no prime factor less th

let n be a positive integer, if n has no prime factor less than or equal to sqt(n), prove that n is prime.

(I need the answer in details please!)

Solution

Suppose n is a positive integer such that n=p*q
here p and q are prime numbers.

Assume that
---> p>n and
---> q>n.

if we multiply these inequalities we will have
p*q>nn
= p*q>n.

But we assumed that p*q= n
So this is a cointradition
Hence we can conclude that either pn or qn

So for a number to ne non prime there must be a prime factor less than or equal to n

Since in our question there is no prime factoe less than or equal to n, we can say that n is prime number