Primitive roots

luckily, $2$ is a primitive root. One needs to check $2^i \not\equiv 1 \ mod \ 29$ for $1\leq i \leq 14$ to know that $2$ is actually a primitive root. After that, we know that numbers of form $2^j \ mod \ 29$ , such that $gcd(j,\phi(29))=gcd(j,28)=1$ and $1 \leq j \leq 27$ , are our primitive roots. There would be $\phi(28)=12$ primitive roots and

$S=(2)+(2^3)+(2^5)+(2^9)+(2^{11})+(2^{13})+(2^{15})+(2^{17})+(2^{19})+(2^{23})+(2^{25})+(2^{27})=2+8+3+19+18+14+27+21+26+10+11+15=174$

here $(.)$ is a notation for computing the argument modulo $29$ .