A school has A students and B teachers. If A < 150, B < 25, and classes have a maximum of 15 students, can the A students be distributed among the B teachers so that each class has the same number of students?
(Assume that any student can be taught by any teacher, and any class has exactly one teacher associated with it.)
(1) It is possible to divide the students evenly into groups of 2, 3, 5, 6, 9, 10, or 15.
(2) The greatest common factor of a and b is 10.
\[2 \leqslant A \leqslant 149\,\,\,\operatorname{int} \,\,\,\,\left( * \right)\,\,\,\,\left( {A\,\, = \,\,\# \,\,{\text{students}}} \right)\]
\[2 \leqslant B \leqslant 24\,\,\,\operatorname{int} \,\,\,\,\,\,\,\,\,\left( {B\,\, = \,\,\# \,\,{\text{classes}}\,\,{\text{ = }}\,\,\,\# \,\,{\text{teachers}}} \right)\]
\[\left( {\frac{{n\,\,{\text{students}}}}{{1\,\,\,{\text{class}}}}} \right)\,\,\,\left( {B\,\,{\text{classes}}} \right)\,\,\,\mathop = \limits^? \,\,A\,\,\,\mathop \Leftrightarrow \limits^{1\,\, \leqslant \,\,n\,\,\operatorname{int} \,\, \leqslant \,\,15} \,\,\,\,\,\boxed{\,\,?\,\,\,\,:\,\,\,\,1 \leqslant \,\,\frac{A}{B}\,\,\, = \,\,\operatorname{int} \,\, \leqslant 15\,\,\,}\]
\[\left( 1 \right) \cap \left( * \right)\,\,\,\left\{ \begin{gathered}
A\,\,{\text{is}}\,\,{\text{a}}\,\,{\text{multiple}}\,\,{\text{of}}\,\,LCM\left( {2,3,5,6,9,10,15} \right) = 90 \hfill \\
\left( * \right)\,\,\,\,2 \leqslant A \leqslant 149 \hfill \\
\end{gathered} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,A = 90\]
\[\left( {1 + 2} \right) \cap \left( * \right)\,\,\,\left\{ \begin{gathered}
A = 90\,\,\,\,\,;\,\,\,\,\,2 \leqslant B \leqslant 24\,\,\,\operatorname{int} \,\,\, \hfill \\
GCD\left( {A,B} \right) = 10 \hfill \\
\end{gathered} \right.\]
\[\left\{ \begin{gathered}
\,{\text{Take}}\,\,\left( {{\text{A,B}}} \right) = \left( {90,10} \right)\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\,\,\,\,\,\,\left( {\frac{{90}}{{10}} = 9} \right) \hfill \\
\,{\text{Take}}\,\,\left( {{\text{A,B}}} \right) = \left( {90,20} \right)\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\,\,\,\,\,\,\,\left( {\frac{{90}}{{20}} \ne \operatorname{int} } \right) \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,{\text{INSUFF}}.\]
The right answer is therefore [spoiler]__(E)__[/spoiler].
This solution follows the notations and rationale taught in the GMATH method.
Regards,
fskilnik.
