Từ \(8b-9a=31\Leftrightarrow8b=9a+31\)
Ta có: \(\dfrac{11}{17}< \dfrac{a}{b}< \dfrac{23}{29}\Rightarrow\left\{{}\begin{matrix}17a>11b\\29a< 23b\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}17.8a>11.8b\\29.8a< 23.8b\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}136a>11\left(9a+31\right)\\232a< 23\left(9a+31\right)\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}136a>99a+341\\232a< 207a+713\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}37a>341\\25a< 713\end{matrix}\right.\)
\(\Rightarrow\dfrac{341}{37}< a< \dfrac{713}{25}\)
Mà a là số tự nhiên \(\Rightarrow9< a< 29\) (1)
Lại có \(8b-9a=31\Leftrightarrow8\left(b-a\right)=a+31\)
\(\Rightarrow a+31\) chia hết cho 8 \(\Rightarrow a\) chia 8 dư 1 (2)
(1);(2) \(\Rightarrow\left[{}\begin{matrix}a=17\\a=25\end{matrix}\right.\)
Với \(a=17\Rightarrow b=23\)
Với \(a=25\Rightarrow b=32\)
