If a<b<0 are integers, what is the value of a^3-b^3?

fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 16)

$$? = {a^3} - {b^3}$$
$$\,a < b < 0\,\,\,{\rm{ints}}\,\,\,\,\left( * \right)$$

$$\left( 1 \right)\,\,\,\left\{ \matrix{
\,\left( {a + b} \right)\left( {a - b} \right) = 65 = 1 \cdot 65 = 5 \cdot 13 \hfill \cr
\,\left( * \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,a + b < a - b\,\,\, < 0\,\,\,{\rm{ints}} \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\rm{I}}.\,\,\left\{ \matrix{
\,a + b = - 65 \hfill \cr
\,a - b = - 1 \hfill \cr} \right.\,\,\,\,\,\,\,\,{\rm{or}}\,\,\,\,\,\,{\rm{II}}.\,\,\left\{ \matrix{
\,a + b = - 13 \hfill \cr
\,a - b = - 5 \hfill \cr} \right.$$
$${\rm{I}}.\,\,\left\{ \matrix{
\,a + b = - 65 \hfill \cr
\,a - b = - 1 \hfill \cr} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,\, \ldots \,\,\,\,\, \Rightarrow \,\,\,\,\left( {a,b} \right) = \left( { - 33, - 32} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = {\left( { - 33} \right)^3} - {\left( { - 32} \right)^3}$$
$${\rm{II}}.\,\,\left\{ \matrix{
\,a + b = - 13 \hfill \cr
\,a - b = - 5 \hfill \cr} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,\, \ldots \,\,\,\,\, \Rightarrow \,\,\,\,\left( {a,b} \right) = \left( { - 9, - 4} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = {\left( { - 9} \right)^3} - {\left( { - 4} \right)^3} \ne {\left( { - 33} \right)^3} - {\left( { - 32} \right)^3}$$

$$\left( 2 \right)\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( { - 2, - 1} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\left( { - 2} \right)^{\rm{3}}} - {\left( { - 1} \right)^3}\,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( { - 3, - 1} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\, = {\left( { - 3} \right)^{\rm{3}}} - {\left( { - 1} \right)^3}\,\, \ne \,{\left( { - 2} \right)^{\rm{3}}} - {\left( { - 1} \right)^3}\,\, \hfill \cr} \right.$$

$$\left( {1 + 2} \right)\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{II}}.\,\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{SUFF}}.\,\,\,\,$$

The correct answer is (C).

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.

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