$\rm\begin{eqnarray}{\bf Hint}\rm\quad a+13b &=&\rm 11m &\:\Rightarrow\:&\rm 13a+169b &=&\rm 143m\\ \rm a+11b &=&\rm 13n &\:\Rightarrow\:&\rm 11a+121b &=&\rm 143n\\ && &\:\Rightarrow\:&\rm\ \ 2a+\ \ 48b &=&\rm 143(m-n)\ \ \ by\ subtracting\ prior\ from\ first \end{eqnarray}$
Thus $\rm\:a + 24b = 143c.\:$ Its solution with minimal $\rm\:a+b\:$ is $\rm\:(a,b) = (23,5),\:$ since all other solutions arise by adding to $\rm\:(a,b)\:$ nonnegative multiples of $\rm\:(-1,6)\:$ or $\rm\:(24,-1),\:$ increasing $\rm\:a+b,$ viz.
$\begin{eqnarray}\rm a+24b = 143&\:\Rightarrow\:&\rm (a,b) = (23,\ \ 5),\ (47,\ \ 4),\ (71,\ \ 3),\ \ldots\\ \rm a+24b = 286&\:\Rightarrow\:&\rm (a,b) = (22,11),\ (46,10),\ (70,\ \ 9),\ \ldots\\ \rm a+24b = 429&\:\Rightarrow\:&\rm (a,b) = (21,17),\ (45,16),\ (69,15),\ \ldots\\ \cdots && \cdots \end{eqnarray}$ In this solution table, adding $\rm\:(24,-1)\:$ moves right, and adding $\rm\:(-1,6)\:$ moves down. But moving left or above the border by subtracting these terms results in a solution with $\rm\:a\:$ or $\rm\:b\:$ negative.