BTGmoderatorLU wrote:Source: Economist GMAT
When m is divided by 7, the remainder is 5. When m is divided by 13, the remainder is 6. If 1 < m < 200, what is the greatest possible value of m?
A. 5
B. 19
C. 61
D. 74
E. 110
\[? = {m_{\,\max }}\,\,\,\left( {1 < m < 200} \right)\,\,{\text{such}}\,\,{\text{that}}\,\,\,\,\left\{ \begin{gathered}
m = 7Q + 5\,\,\,\left( 1 \right)\,\, \hfill \\
m = 13K + 6\,\,\,\left( 2 \right) \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\left( {Q,K\,\,\,{\text{ints}}} \right)\]
\[\left\{ \begin{gathered}
\left( 1 \right) \cdot 13\,\,\,\, \Rightarrow \,\,\,\,13m = 7 \cdot 13Q + 65 \hfill \\
\left( 2 \right) \cdot 7\,\,\,\,\,\, \Rightarrow \,\,\,\,7m = 7 \cdot 13K + 42 \hfill \\
\end{gathered} \right.\,\,\,\,\mathop \Rightarrow \limits^{\left( - \right)} \,\,\,\,6m = 7 \cdot 13\left( {Q - K} \right) + 23\]
\[\left\{ \begin{gathered}
\,7m = 7 \cdot 13K + 42 \hfill \\
\,6m = 7 \cdot 13\left( {Q - K} \right) + 23 \hfill \\
\end{gathered} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{\left( - \right)} \,\,\,\,m = 7 \cdot 13\left( {2K - Q} \right) + 19\]
\[\left\{ \begin{gathered}
\,m = 91J + 19 \hfill \\
\,1 < m < 200 \hfill \\
\end{gathered} \right.\,\,\,\,\left( {J\,\,\operatorname{int} } \right)\,\,\,\, \Rightarrow \,\,\,\,\,{J_{\max }} = 1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = {m_{\,\max }} = 91 + 19 = 110\]
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio. 
