I'm trying to generate $U_{10}$ (Euler group; multiplicative group of units modulo $10$) with $3$.
If $U_{10}=\{1,3,7,9\}$, let's try to generate all other elements using $3$:
$\begin{align*} \langle 3\rangle = 3 &\longrightarrow 3+3=6\\ &\longrightarrow 6+3=9\\ &\longrightarrow 9+3=12\\ &\longrightarrow 12\bmod 10 = 2\\ &\longrightarrow 2+3 = 5\\ &\longrightarrow 5+3 = 8\\ &\longrightarrow 8+3 =11\\ &\longrightarrow 11\bmod 10 = 1\\ &\longrightarrow 1+3 = 4\\ &\longrightarrow 4+3 = 7\\ &\longrightarrow \text{Done.} \end{align*}$
Is this correct ?
Regards,