Data Sufficiency

[Math Revolution GMAT math practice question]

If n is an integer between 30 and 50 inclusive, what is the value of n?

1) When n is divided by 8, the remainder is 7
2) When n is divided by 16, the remainder is 7

Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If n is an integer between 30 and 50 inclusive, what is the value of n?

1) When n is divided by 8, the remainder is 7
2) When n is divided by 16, the remainder is 7

$$30\,\,\, \le \,\,\,n\,\,{\mathop{\rm int}} \,\,\, \le \,\,\,50\,\,\,\,\left( * \right)$$
$$? = n$$

$$\left( 1 \right)\,\,n = 8Q + 7\,\,,\,\,Q\,\,{\mathop{\rm int}} \,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,Q = 3\,\,\,\, \Rightarrow \,\,\,\,\,? = 31\,\, \hfill \cr
\,{\rm{Take}}\,\,Q = 4\,\,\,\, \Rightarrow \,\,\,\,\,? = 39\,\, \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{INSUFF}}.$$

$$\left( 2 \right)\,\,n = 16K + 7,\,\,K\,\,{\mathop{\rm int}} \,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,K = 2\,\,\,\,\,\, \Rightarrow \,\,\,\,? = 39\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}{\rm{.}}$$
$$\left( {**} \right)\,\,\,39 - 16 < 30\,\,\,{\rm{and}}\,\,\,39 + 16 > 50\,\,,\,\,\,{\rm{impossible}}\,\,{\rm{by}}\,\,\left( * \right)$$


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

Regards,
Fabio.

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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, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
We can express n = 8k+7 for some integer k.
If k = 3, then n = 31.
If k = 4, then n = 39.
Since we don't have a unique solution, condition 1) is not sufficient.

Condition 2)
We can express n = 16m+7 for some integer m.
If m = 2, then n = 39.
If m = 1, then n = 23 and n < 30.
If m = 3, then n = 55 and n > 50.
Thus n = 39 is the unique solution and condition 2) is sufficient.

Therefore, B is the answer.
Answer: B

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.