A school has a students and b teachers. If a < 150, b < 25, and classe

\(2 \leqslant A \leqslant 149\,\,\,\operatorname{int} \,\,\,\,\left( * \right)\,\,\,\,\left( {A\,\, = \,\,\# \,\,{\text{students}}} \right)\)

\(2 \leqslant B \leqslant 24\,\,\,\operatorname{int} \,\,\,\,\,\,\,\,\,\left( {B\,\, = \,\,\# \,\,{\text{classes}}\,\,{\text{ = }}\,\,\,\# \,\,{\text{teachers}}} \right)\)

\(\left( {\frac{{n\,\,{\text{students}}}}{{1\,\,\,{\text{class}}}}} \right)\,\,\,\left( {B\,\,{\text{classes}}} \right)\,\,\,\mathop = \limits^? \,\,A\,\,\,\mathop \Leftrightarrow \limits^{1\,\, \leqslant \,\,n\,\,\operatorname{int} \,\, \leqslant \,\,15} \,\,\,\,\,\boxed{\,\,?\,\,\,\,:\,\,\,\,1 \leqslant \,\,\frac{A}{B}\,\,\, = \,\,\operatorname{int} \,\, \leqslant 15\,\,\,}\)

\(\left( 1 \right) \cap \left( * \right)\,\,\,\left\{ \begin{gathered}\\
A\,\,{\text{is}}\,\,{\text{a}}\,\,{\text{multiple}}\,\,{\text{of}}\,\,LCM\left( {2,3,5,6,9,10,15} \right) = 90 \hfill \\\\
\left( * \right)\,\,\,\,2 \leqslant A \leqslant 149 \hfill \\ \\
\end{gathered} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,A = 90\)

\(\left( {1 + 2} \right) \cap \left( * \right)\,\,\,\left\{ \begin{gathered}\\
A = 90\,\,\,\,\,;\,\,\,\,\,2 \leqslant B \leqslant 24\,\,\,\operatorname{int} \,\,\, \hfill \\\\
GCD\left( {A,B} \right) = 10 \hfill \\ \\
\end{gathered} \right.\)

\(\left\{ \begin{gathered}\\
\,{\text{Take}}\,\,\left( {{\text{A,B}}} \right) = \left( {90,10} \right)\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\,\,\,\,\,\,\left( {\frac{{90}}{{10}} = 9} \right) \hfill \\\\
\,{\text{Take}}\,\,\left( {{\text{A,B}}} \right) = \left( {90,20} \right)\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\,\,\,\,\,\,\,\left( {\frac{{90}}{{20}} \ne \operatorname{int} } \right) \hfill \\ \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,{\text{INSUFF}}.\)


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

Regards,
fskilnik.