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.
The OA is the option C.
Why is none of the statement sufficient alone? I would be thankful if someone explains this question to me. <i class="em em-smiley"></i>
If p and q are distinct integers, is 4 a factor of
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Try TESTING VALUES to prove each statement insufficient.VJesus12 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.
The OA is the option C.
Why is none of the statement sufficient alone? I would be thankful if someone explains this question to me. <i class="em em-smiley"></i>
(1) 4 is a factor of p.
Say that p = 12. Since no information is given about q, q could be anything:
- If q = 7, then p - q = 5, so the answer is no, 4 is not a factor of p - q.
- If q = 8, then p - q = 4, so the answer is yes, 4 is a factor of p - q.
Since we found values that could give a "yes" or "no" answer, this statement is insufficient.
(2) 4 is a factor of q.
Say that q = 8. Since no information is given about p, p could be anything:
- If p = 11, then p - q = 3, so the answer is no, 4 is not a factor of p - q.
- If p = 16, then p - q = 8, so the answer is yes, 4 is a factor of p - q.
Since we found values that could give a "yes" or "no" answer, this statement is insufficient.
(1) & (2) together:
If p and q are both multiples of 4, then p - q MUST also be a multiple of 4:
Sufficient. The answer is C.RULE: any multiple of a given integer n plus or minus another multiple of n will equal a multiple of n.
Ceilidh Erickson
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Harvard Graduate School of Education
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\[p \ne q\,\,\,{\text{ints}}\]VJesus12 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.
\[\frac{{p - q}}{4}\,\,\,\mathop = \limits^? \,\,\,\operatorname{int} \]
\[\left( 1 \right)\,\,\,\,\frac{p}{4} = \operatorname{int} \,\,\,\,\left\{ \begin{gathered}
\,\,Take\,\,\left( {p,q} \right) = \left( {0,1} \right)\,\,\,\, \Rightarrow \,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\,\,Take\,\,\left( {p,q} \right) = \left( {0,4} \right)\,\,\,\, \Rightarrow \,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right)\,\,\,\,\frac{q}{4} = \operatorname{int} \,\,\,\,\left\{ \begin{gathered}
\,\,Take\,\,\left( {p,q} \right) = \left( {1,0} \right)\,\,\,\, \Rightarrow \,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\,\,Take\,\,\left( {p,q} \right) = \left( {4,0} \right)\,\,\,\, \Rightarrow \,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\]
\[\left( {1 + 2} \right)\,\,\,\,\frac{{p - q}}{4} = \frac{p}{4} - \frac{q}{4} = \operatorname{int} - \operatorname{int} = \operatorname{int} \,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\]
The above follows the notations and rationale taught in the GMATH method.
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