AAPL wrote:Manhattan Prep
If Polygon X has fewer than 9 sides, how many sides does Polygon X have?
1) The sum of the interior angles of Polygon X is divisible by 16.
2) The sum of the interior angles of Polygon X is divisible by 15.
$$\sum\nolimits_N { = \left( {N - 2} \right) \cdot 180\,\,\,\,\,\,\left[ {{\rm{degrees}}} \right]} $$
$$? = N\,\,\,\,\left( {3 \le \,\,N\,\,{\mathop{\rm int}} \,\, \le 8} \right)$$
$$\left( 1 \right)\,\,\,{\mathop{\rm int}} \,\, = \,\,{{\left( {N - 2} \right) \cdot 180} \over {16}}\,\, = \,\,{{\left( {N - 2} \right) \cdot 45} \over 4}\,\,\,\,\,\mathop \Rightarrow \limits^{GCF\left( {45,4} \right)\, = \,1} \,\,\,\,\,{{N - 2} \over 4} = {\mathop{\rm int}} $$
$$\left. \matrix{
3 \le \,\,N\,\,{\mathop{\rm int}} \,\, \le 8\,\, \hfill \cr
{{N - 2} \over 4} = {\mathop{\rm int}} \hfill \cr} \right\}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,N = 6\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.$$
$$\left( 2 \right)\,\,\,{\mathop{\rm int}} \,\, = \,\,{{\left( {N - 2} \right) \cdot 180} \over {15}}\,\, = \,\,\left( {N - 2} \right) \cdot 12\,\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,N = 3 \hfill \cr
\,{\rm{Take}}\,\,N = 4 \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{INSUFF}}.$$
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
