<s>[INST] Find the smallest positive integer that satisfies the system of congruences $N \equiv 2 \pmod{11}, N \equiv 3 \pmod{17}. $ 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!
 [/INST] 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
$$ \color{red}{N}=17\cdot 9+3 = \color{red}{156}.$$
</s>