Find the smallest positive integer that is greater than 1 and relatively prime to the product of the first 20 positive integers. Reminder: two numbers are relatively prime if their greatest common divisor is 1.

We have this set of integers: {1,2,3,...,18,19,20} in which we multiply them all to get some very massive number, which I'll call M. The actual number itself doesn't matter.

The goal is to find the smallest number x such that x has no factors in common with M (other than 1).

Trying anything from x = 2 to x = 20 won't work as those values themselves are in the set mentioned.
If we tried x = 21, then it won't work because 21 = 3*7. Both 3 and 7 are in the set listed above. In other words, x and M have the shared factors 3 and 7 which means x and M are not relatively prime.
Trying x = 22 doesn't work either because 22 = 2*11. Both 2 and 11 are in that set as well. The x and M values aren't relatively prime.
Luckily, x = 23 does work. This is because 23 only has the factors 1 and itself. It's a prime number. So that means 23 and this massive number have no factors in common other than 1. Therefore, the GCD of 23 and this massive number is 1. By definition, it indicates these two values are relatively prime or coprime.

In short, we simply go for the smallest prime that's larger than 20.