##### The Idea

A number is said to be prime if the only natural numbers that divide itself are 1 and itself. By the Fundamental Theorem of Arithmetic, every number has a unique representation as a product of primes (up to permutations of the prime products). Therefore, as long as 600,851,475,143 is not prime,

at least all but one of its prime divisors must be less than

$\sqrt{600,851,475,143}$. That is, if we keep dividing the target of 600,851,475,143 by its prime divisors that are less than its square root, we will either get that the left over is greater than the max divisor we divided by and it is prime, or the max prime divisor found in the loop is the actual max divisor.