Условие:

К числу 9 справа и слева припиши одну и ту же такую цифру, чтобы полученное трехзначное число делилось на 7 без остатка.

Найдем число путем перебора:
$\begin{array}{cc}\begin{array}{l}\underset{¯}{\phantom{0}}191\\ \phantom{0}\underset{¯}{14}\\ \phantom{0}\underset{¯}{\phantom{0}}51\\ \phantom{00}\underset{¯}{49}\\ \phantom{000}2\end{array}& \begin{array}{l}\overline{)7}\\ \overline{\overline{)27}}\end{array}\end{array}$
 
$\begin{array}{cc}\begin{array}{l}\underset{¯}{\phantom{0}}292\\ \phantom{0}\underset{¯}{28}\\ \phantom{0}\underset{¯}{\phantom{0}}12\\ \phantom{000}\underset{¯}{7}\\ \phantom{000}5\end{array}& \begin{array}{l}\overline{)7}\\ \overline{\overline{)41}}\end{array}\end{array}$
 
$\begin{array}{cc}\begin{array}{l}\underset{¯}{\phantom{0}}393\\ \phantom{0}\underset{¯}{35}\\ \phantom{0}\underset{¯}{\phantom{0}}43\\ \phantom{00}\underset{¯}{42}\\ \phantom{000}1\end{array}& \begin{array}{l}\overline{)7}\\ \overline{\overline{)56}}\end{array}\end{array}$
 
$\begin{array}{cc}\begin{array}{l}\underset{¯}{\phantom{0}}494\\ \phantom{0}\underset{¯}{49}\\ \phantom{00}\underset{¯}{\phantom{0}}4\\ \phantom{000}\underset{¯}{0}\\ \phantom{000}4\end{array}& \begin{array}{l}\overline{)7}\\ \overline{\overline{)70}}\end{array}\end{array}$
 
$\begin{array}{cc}\begin{array}{l}\underset{¯}{\phantom{0}}595\\ \phantom{0}\underset{¯}{56}\\ \phantom{0}\underset{¯}{\phantom{0}}35\\ \phantom{00}\underset{¯}{35}\\ \phantom{000}0\end{array}& \begin{array}{l}\overline{)7}\\ \overline{\overline{)85}}\end{array}\end{array}$
 
$\begin{array}{cc}\begin{array}{l}\underset{¯}{\phantom{0}}696\\ \phantom{0}\underset{¯}{63}\\ \phantom{0}\underset{¯}{\phantom{0}}66\\ \phantom{00}\underset{¯}{63}\\ \phantom{000}3\end{array}& \begin{array}{l}\overline{)7}\\ \overline{\overline{)99}}\end{array}\end{array}$
 
$\begin{array}{cc}\begin{array}{l}\underset{¯}{\phantom{0}}797\\ \phantom{0}\underset{¯}{7}\\ \phantom{0}\underset{¯}{\phantom{0}}9\\ \phantom{00}\underset{¯}{7}\\ \phantom{0}\underset{¯}{\phantom{0}}27\\ \phantom{00}\underset{¯}{21}\\ \phantom{000}6\end{array}& \begin{array}{l}\overline{)7}\\ \overline{\overline{)113}}\end{array}\end{array}$
 
$\begin{array}{cc}\begin{array}{l}\underset{¯}{\phantom{0}}898\\ \phantom{0}\underset{¯}{7}\\ \underset{¯}{\phantom{0}}19\\ \phantom{0}\underset{¯}{14}\\ \phantom{0}\underset{¯}{\phantom{0}}58\\ \phantom{00}\underset{¯}{56}\\ \phantom{000}2\end{array}& \begin{array}{l}\overline{)7}\\ \overline{\overline{)128}}\end{array}\end{array}$
 
$\begin{array}{cc}\begin{array}{l}\underset{¯}{\phantom{0}}999\\ \phantom{0}\underset{¯}{7}\\ \underset{¯}{\phantom{0}}29\\ \phantom{0}\underset{¯}{28}\\ \phantom{0}\underset{¯}{\phantom{0}}19\\ \phantom{00}\underset{¯}{14}\\ \phantom{000}5\end{array}& \begin{array}{l}\overline{)7}\\ \overline{\overline{)142}}\end{array}\end{array}$
 
Ответ: 595