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

Let n be non-prime. Therefore, it has at least two integer factors greater than 1. Let f be the smallest of n's such factors. Suppose f > sqrt n. Then n/f is an integer LTE sqrt n, thus smaller than f. Therefore, f cannot be n's smallest factor. Reductio ad absurdum; n's smallest factor must be LTE sqrt n.