Find a fraction $\frac{m}{n}$ which satisfies the given condition Find a fraction such that all of $\frac{m}{n}$, $\frac{m+1}{n+1}$, $\frac{m+2}{n+2}$, $\frac{m+3}{n+3}$, $\frac{m+4}{n+4}$, $\frac{m+5}{n+5}$ are reducible by cancellation. Condition: $m≠n$.
What I tried was... I wrote $$\frac{m}{n}=k$$ Then, I replaced $m$ in the fractions as $nk$. And after a bit of simple manipulation, I obtained $$k+\frac{1-k}{n+1},$$ $$k+\frac{2-k}{n+2},$$ $$k+\frac{3-k}{n+3},$$ $$k+\frac{4-k}{n+4},$$ $$k+\frac{5-k}{n+5}$$ Now I do not know how to proceed any further.

Simple computer search gives:
\begin{array}{|c|c|} \hline
  m & n \\ \hline
  212 & 2 \\
  213 & 3 \\
  214 & 4 \\
  215 & 5 \\
  300 & 90 \\
  301 & 91 \\
  324 & 114 \\
  325 & 115 \\ \hline
\end{array}
Further observation:
\begin{array}{|c|c|} \hline
  n \equiv m \pmod{\lambda} & \lambda \\ \hline
  2,3,4,5 & 210 \\
  6,7 & 462 \\
  8,9,10,11 & 858 \\
  10,11 & 1430 \\
  12,13 & 1326 \\
  12,13 & 2210 \\
  14,15,16,17 & 1938 \\
  14,15 & 3230 \\
  16,17 & 4522 \\
  18,19 & 2622 \\
  18,19 & 6118 \\
  20,21,22,23 & 690 \\
  20,21 & 1610 \\
  20 & 3795 \\ 
  24,25 & 870 \\ \hline
\end{array}
