What is a-b?
(1) a^3 - b^3 = 117
(2) a^2 + ab + b^2 = 39
OA C
Source: Veritas Prep
What is a–b?
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Note that a^3 - b^3 = (a - b) (a^2 + ab + b^2).BTGmoderatorDC wrote:What is a-b?
(1) a^3 - b^3 = 117
(2) a^2 + ab + b^2 = 39
OA C
Source: Veritas Prep
=> a - b = (a^3 - b^3) / (a^2 + ab + b^2)
Since each statement alone cannot return the unique value of (a - b), we need the values of (a^3 - b^3) and (a^2 + ab + b^2).
Thus, a - b = (a^3 - b^3) / (a^2 + ab + b^2) = 117/39 = 3. Sufficient.
The correct answer: C
Hope this helps!
-Jay
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\[? = a - b\]BTGmoderatorDC wrote:What is a-b?
(1) a^3 - b^3 = 117
(2) a^2 + ab + b^2 = 39
Source: Veritas Prep
\[\left( 1 \right)\,\,{a^3} - {b^3} = 117\,\,\,\left\{ \begin{gathered}
\,\,{\text{Take}}\,\,\,\left( {a;b} \right) = \left( {\sqrt[{3\,}]{{117}}\,\,;\,\,0} \right)\,\,\,\,\, \Rightarrow \,\,\,\,? = \sqrt[{3\,}]{{117}}\,\,\, \hfill \\
\,\,{\text{Take}}\,\,\,\left( {a;b} \right) = \left( {1\,\,;\,\, - \sqrt[{3\,}]{{116}}} \right)\,\,\,\,\, \Rightarrow \,\,\,\,? = 1 + \sqrt[{3\,}]{{116}}\,\,\, \ne \sqrt[{3\,}]{{117}}\,\, \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right)\,\,{a^2} + ab + {b^2} = 39\]
\[\left\{ \begin{gathered}
\,\,{\text{Take}}\,\,\,\left( {a;b} \right) = \left( {\sqrt {39} \,\,;\,\,0} \right)\,\,\,\,\, \Rightarrow \,\,\,\,? = \sqrt {39} \,\,\, \hfill \\
\,\,{\text{Take}}\,\,\,\left( {a;b} \right) = \left( {0\,\,;\,\,\sqrt {39} } \right)\,\,\,\,\, \Rightarrow \,\,\,\,? = - \sqrt {39} \,\,\, \ne \sqrt {39} \,\, \hfill \\
\end{gathered} \right.\,\,\,\]
\[\left( {1 + 2} \right)\,\,\,\left\{ \begin{gathered}
\,\,\left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right) = {a^3} - {b^3} = 117 \hfill \\
\,\,{a^2} + ab + {b^2} = 39 \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,39\left( {a - b} \right) = 117\,\,\,\,\, \Rightarrow \,\,? = a - b\,\,\,{\text{unique}}\,\,\]
This solution follows the notations and rationale taught in the GMATH method.
Regards,
fskilnik.
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