BTGmoderatorDC wrote:If the positive integer n is greater than 6, what is the remainder when n is divided by 6?

(1) When n is divided by 9, the remainder is 2.
(2) When n is divided by 4, the remainder is 1.
Source: Manhattan Prep

\[n \geqslant 7\,\,\,\operatorname{int} \,\,\,\,\left( * \right)\]
$$\left. \matrix{
n = 6Q + R\,\,\, \hfill \cr
Q\,\,\mathop \ge \limits^{\left( * \right)} \,\,1\,\,{\mathop{\rm int}} \hfill \cr} \right\}\,\,\,\,\,? = R\,\,\,\,\,\,\left( {0 \le R \le 5\,\,{\mathop{\rm int}} } \right)$$
\[\left( 1 \right)\,\,\,n = 9K + 2\,\,\,,\,\,\,K\,\mathop \geqslant \limits^{\left( * \right)} \,1\,\,\operatorname{int} \,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,K = 1\,\,\,\, \Rightarrow \,\,\,n = 11\,\,\,\,\, \Rightarrow \,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{5}}\,\,\,\,\,\left( {Q = 1} \right)\, \hfill \\
\,{\text{Take}}\,\,K = 2\,\,\,\, \Rightarrow \,\,\,n = 20\,\,\,\,\, \Rightarrow \,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{2}}\,\,\,\,\left( {Q = 3} \right) \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right)\,\,\,n = 4J + 1\,\,\,,\,\,\,J\,\mathop \geqslant \limits^{\left( * \right)} \,2\,\,\operatorname{int} \,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,J = 2\,\,\,\, \Rightarrow \,\,\,n = 9\,\,\,\,\, \Rightarrow \,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{3}}\,\,\,\,\,\left( {Q = 1} \right)\, \hfill \\
\,{\text{Take}}\,\,J = 3\,\,\,\, \Rightarrow \,\,\,n = 13\,\,\,\,\, \Rightarrow \,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{1}}\,\,\,\,\left( {Q = 2} \right) \hfill \\
\end{gathered} \right.\]
$$\left( {1 + 2} \right)\,\,\,\,9K = n - 2\,\,\, = \,\,4J - 1\,\,\,\,\, \Rightarrow \,\,\,\,\,n - 2\,\,\,\,\left\{ \matrix{
\,{\rm{odd}}\,\,\,\, \Rightarrow \,\,\,n = {\rm{odd}}\,\,\,\, \Rightarrow \,\,\,R \in \left\{ {1,3,5} \right\} \hfill \cr
\,{\rm{multiple}}\,\,{\rm{of}}\,\,9,\,\,{\rm{hence}}\,\,{\rm{of}}\,\,3\,\,\,\, \Rightarrow \,\,\,n = 3M + 2 \hfill \cr} \right.\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\,\,\,? = 5$$
$$\left( {**} \right)\,\,\,\,\left\{ \matrix{
\,R = 1\,\,\,\, \Rightarrow \,\,\,\,n = 6Q + 1\,\,\,\,\, \Rightarrow \,\,\,\,\,n = 3\left( {2Q} \right) + 1\,\,\,\,\, \Rightarrow \,\,\,\,\,n = 3M + 2\,\,\,\,\,{\rm{impossible}} \hfill \cr
\,R = 3\,\,\,\, \Rightarrow \,\,\,\,n = 6Q + 3\,\,\,\,\, \Rightarrow \,\,\,\,\,n = 3\left( {2Q + 1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,n = 3M + 2\,\,\,\,\,{\rm{impossible}} \hfill \cr} \right.\,\,\,\,\,\,\,\,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.