I'm working on a sample problem where I'm figuring out if you can make primes concatenating the numbers 1-5 in any order, just that they all be used.
I first tried to look at the combinations of the last digits of the number to see if I could show that no primes are possible. For example, I know that the number can't end in 2,4,or 5 because that number would clearly be composite. So to end the number in either 1 or 3, then these are the only possibilities for primes. I thought that maybe I look at modulus, such as $$x ≡ 3 \pmod{10}.$$ But from Dirichlet's theorem there are infinite primes of the form $10n + 3$, so I can't rule that out either.
This is my attempt to practice for a much larger problem, so if the example of $1-5$ doesn't work very well please let me know.