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

So to check whether a number N is Prime or not. We need to only check if N is divisible by numbers<=SQROOT(N). This is because, if we factor N into any 2 factors say X and Y, ie. N=XY. Each of X and Y cannot be less than SQROOT(N) because then, XY < N Each of X and Y cannot be greater than SQROOT(N) because then, X*Y > N

Therefore one factor must be less than or equal to SQROOT(N) ( while the other factor is greater than or equal to SQROOT(N) ). So to check if N is Prime we need only check those numbers <= SQROOT(N).