Near the end of Euclid's proof he says that if you multiply all our known primes from 2 to Pn and then add 1 to it, the number isn't divisible by any of our primes up to Pn, because it leaves the remainder 1. Why is that? How does he know that it will always leave the remainder 1? Couldn't adding 1 just make a slightly larger number that has a factor that might be 2 or Pn? If someone can set my stoopid brain straight I'd be very happy. Thanks.