Determine $p$ for one to have $\displaystyle\frac{1}{5}+\frac{1}{6}=\frac{1+1}{5+6}=\frac{2}{11}$, in $\mathbb{Z_p}$.
Try multiplying both sides by $11 \cdot 5 \cdot 6$, to get $$121 = 11 \cdot 6 + 11 \cdot 5 = 2 \cdot 5 \cdot 6 = 60$$ in $\mathbb{Z}/p\mathbb{Z}$.
In addition to $p=121-60=61$ we can use the extended Euclidean algorithm to obtain $5^{-1}=-12$, $6^{-1}=-10$ and $11^{-1}=-11$ in $\mathbb{Z}/61 \mathbb{Z}$. Hence $$ \frac{1}{5}+\frac{1}{6}=-12-10=-22=2\cdot (-11)=\frac{2}{11}. $$