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 :: http://www.GMATH.net (Math for the GMAT)

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