Solution: We will solve this with two methods; the first one is by substitution and other one is by using the Chinese Remainder theorem. Let $x = 7k + 6$. We want \begin{align*} 7k + 6 \equiv 4 \mod 5 & \implies 2k \equiv -2 \mod 5 \\ & \implies k \equiv -1 \mod 5. \end{align*} Note that $k = 4$ is a solution. Therefore $x = 34$ is a solution satisfying the last two congruent equations. A general solution satisfying the last two congruent equations will be $x = 34 + 35k$. Therefore we have \begin{align*} 35k + 34 \equiv 2 \mod 3 & \implies 2k + 1 \equiv 2 \mod 3 \\ & \implies 2k \equiv 1 \mod 3. \end{align*} Hence, a solution to the given system of congruence equation will be $x = 35\times 2 + 34 = 104$. This will be the least positive number as the solution is unique in $\mathbb{Z} _{105}$.