Why do we test only the prime numbers less than the square root of the given number?

Answer 1

First, you do not. You must also test the square root of the given number. So the question should be in terms of "less than or equal to".

Suppose you wish to test the number n and suppose s is the square root of n.

Then s*s = n

Now suppose p is factor of n, with factor pair q. so that n = p*q and, without loss of generality, assume that p ≤ q.

Thus p*q = s*s so that p ≤ s ≤ q

That is to say, one of the pair of factors of n will be less or equal to its square root while the other member of the factor pair will be greater or equal to the square root.