BTGmoderatorDC wrote:M is a rectangular solid. Find the volume of M
Statement #1: The bottom face of M has an area of 28, and the front face, an area of 35.
Statement #2: All three dimensions of M are positive integers greater than one.
Source: Magoosh
$$? = abc$$
$$\left( 1 \right)\,\,\left( {ab,ac} \right) = \left( {28,35} \right)\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {1,28,35} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 28 \cdot 35 \hfill \cr
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {2,14,{{35} \over 2}} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 2 \cdot 14 \cdot {{35} \over 2} \ne 28 \cdot 35 \hfill \cr} \right.$$
$$\left( 2 \right)\,\,a,b,c\,\, \ge 2\,\,{\rm{ints}}\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {2,2,2} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 2 \cdot 2 \cdot 2 \hfill \cr
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {2,2,3} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 2 \cdot 2 \cdot 3 \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,\,\,\left\{ \matrix{
ac = 35 \hfill \cr
a,c\,\, \ge 2\,\,{\rm{ints}} \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\{ \matrix{
\left( {a,c} \right) = \left( {5,7} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{ab\,\, = \,\,28} \,\,\,\,b = {{28} \over 5}\,\,\,\,{\rm{impossible}}\,\,\,\,\left( {b\,\,{\mathop{\rm int}} } \right) \hfill \cr
\,{\rm{or}}\, \hfill \cr
\,\left( {a,c} \right) = \left( {7,5} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{ab\,\, = \,\,28} \,\,\,\,b = 4\,\,\,\, \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 7 \cdot 4 \cdot 5$$
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
