Why do we check up to the square root of a prime number to determine if it is prime?

Question

To test whether a number is prime or not, why do we have to test whether it is divisible only up to the square root of that number?

Answer 1

Suppose n is not a prime number (greater than 1). So there are numbers a and b such that

n = ab      (1 < a <= b < n)

By multiplying the relation a<=b by a and b we get:

a^2 <= ab
 ab <= b^2

Therefore: (note that n=ab)

a^2 <= n <= b^2

Hence: (Note that a and b are positive)

a <= sqrt(n) <= b

So if a number (greater than 1) is not prime and we test divisibility up to square root of the number, we will find one of the factors.