- Verify that 2 is a primitive root mod 25.
I just want to make sure my understanding of what a primitive root is is clear. So to show my work I calculated 2^1mod25 up to 2^24mod25, and showed that all values 1-24 that are relatively prime with 25, (so excluding 5,10,15 and 20) are represented. Is this the correct way of thinking?
Your understanding is correct, though there are faster ways to prove that.
$2$ is a primitive root $\bmod 25$ iff $20$ is the smaller positive exponent $n$ such that $2^n \equiv 1 \bmod 25$.
Since $a^{20} \equiv 1 \bmod 25$ for all $a$ coprime with $25$, you only need to prove that $2^{10} \equiv -1 \bmod 25$, which is immediate, since $2^{10}=1024$.