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n is a positive integer. Is n divisible by 3?

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by Max@Math Revolution » Mon Dec 03, 2018 11:44 pm

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[Math Revolution GMAT math practice question]

n is a positive integer. Is n divisible by 3?

1) 36/n is divisible by 3
2) 27/n is divisible by 3
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Source: — Data Sufficiency |
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by fskilnik@GMATH » Tue Dec 04, 2018 6:05 pm
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

n is a positive integer. Is n divisible by 3?

1) 36/n is divisible by 3
2) 27/n is divisible by 3
$$n \ge 1\,\,{\mathop{\rm int}} $$
$${n \over 3}\,\,\mathop = \limits^? \,\,{\mathop{\rm int}} $$

$$\left( 1 \right)\,\,\,\left( {{{36} \over n} = {\mathop{\rm int}} \,\,{\rm{and}}} \right)\,\,\,{{12} \over n} = {{36} \over {3n}} = {\mathop{\rm int}} \,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,{\rm{n = 1}}\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr
\,{\rm{Take}}\,\,{\rm{n = 3}}\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr} \right.$$

$$\left( 2 \right)\,\,\,\left( {{{27} \over n} = {\mathop{\rm int}} \,\,{\rm{and}}} \right)\,\,\,{9 \over n} = {{27} \over {3n}} = {\mathop{\rm int}} \,\,\,\left\{ \matrix{
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,{\rm{n = 1}}\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,{\rm{n = 3}}\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr} \right.$$

$$ \Rightarrow \,\,\,\,\left( {\rm{E}} \right)$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Max@Math Revolution » Wed Dec 05, 2018 11:16 pm
=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 1 variable (n) and 0 equations in the original condition, D is most likely to be the answer. So, we should consider each condition on its own first. It is suggested we plug in numbers when solving remainder problems.


Condition 1)
If n = 3, then 36/3 = 12 is divisible by 3, and n is divisible by 3. The answer is 'yes'.
If n = 1, then 36/1 = 36 is divisible by 3, but n is not divisible by 3. The answer is 'no'.
Thus, condition 1) is not sufficient, since it does not yield a unique solution.

Condition 2)
If n = 3, then 27/3 = 9 and n is divisible by 3. The answer is 'yes'.
If n = 1, then 27/1 = 27 and n is not divisible by 3. The answer is 'no'.
Thus, condition 2) is not sufficient, since it does not yield a unique solution.

Conditions 1) & 2)
Even if we consider both conditions together, we still have two possible values of n: n = 1 and 3.
Thus, both conditions together are not sufficient, since they do not yield a unique solution.

Therefore, E is the answer.
Answer: E

If the original condition includes "1 variable", or "2 variables and 1 equation", or "3 variables and 2 equations" etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
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