Finding primitive roots

I'm trying to find the smallest primitive root modulo 23 for the operation of multiplication. Is there any sensible way to approach this without brute force calculation?

More generally is there any particular way in which to go about finding a primitive root modulo n? We proved the Euler-Lagrange-Legendre Theorem that states if p is a prime then there are phi(p-1) primitive roots modulo p but that only proves the existence of a number of primitive roots and doesn't give any way of finding them.

Thank you for any help.

phi(23)=22=2x11
if a does not divide 23 then its order mod 23 divides 22. if the order of a is not 22 then its order mod 23 must divide 2 or 11.
hence if a is primitive root then a^2 or a^11 is not =1 mod 23.

2^11=1 mod 23
3^11=1 mod 23
4^11 =1 mod 23
5^2=2 mod 23 and 5^11=22 mod 23
5 is smallest primitive root mod 23.