Find the smallest positive integer that satisfies the system of congruences
\begin{align*}
N &\equiv 2 \pmod{11}, \\
N &\equiv 3 \pmod{17}.
\end{align*}
The only way I know to solve this problem is by listing it all out, and so far, it's not working. Is there a faster way? Thanks for posting a solution!
Best Answer
By the second constraint $N$ is a number of the form $17k+3$.
We may now impose the first constraint: $$ 17k+3\equiv 6k+3 \equiv 3(2k+1) \equiv 2\pmod{11} $$ leading to $2k+1\equiv 8\pmod{11}$, equivalent to $k\equiv 9\pmod{11}$.
It follows that the smallest positive number fulfilling both constraints is given by $$ \color{red}{N}=17\cdot 9+3 = \color{red}{156}.$$