Bunuel wrote:
If p and q are distinct integers, is 4 a factor of p – q?
(1) 4 is a factor of p.
(2) 4 is a factor of q.
\(p \ne q\,\,\,{\text{ints}}\)
\(\frac{{p - q}}{4}\,\,\,\mathop = \limits^? \,\,\,\operatorname{int}\)
We will prove that each statement ALONE is insufficient to answer the question asked (in a unique way), through what we call an ALGEBRAIC BIFURCATION:
\(\left( 1 \right)\,\,\,\,\frac{p}{4} = \operatorname{int} \,\,\,\,\left\{ \begin{gathered}
\,\,Take\,\,\left( {p,q} \right) = \left( {0,1} \right)\,\,\,\, \Rightarrow \,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\,\,Take\,\,\left( {p,q} \right) = \left( {0,4} \right)\,\,\,\, \Rightarrow \,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\)
\(\left( 2 \right)\,\,\,\,\frac{q}{4} = \operatorname{int} \,\,\,\,\left\{ \begin{gathered}
\,\,Take\,\,\left( {p,q} \right) = \left( {1,0} \right)\,\,\,\, \Rightarrow \,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\,\,Take\,\,\left( {p,q} \right) = \left( {4,0} \right)\,\,\,\, \Rightarrow \,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\)
\(\left( {1 + 2} \right)\,\,\,\,\frac{{p - q}}{4} = \frac{p}{4} - \frac{q}{4} = \operatorname{int} - \operatorname{int} = \operatorname{int} \,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\)
The above follows the notations and rationale taught in the GMATH method.
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Fabio Skilnik ::
GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here:
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