Data Sufficiency

[Math Revolution GMAT math practice question]

What is the value of A?

1) The four-digit number A77A is a multiple of 4.
2) The four-digit number A77A is a multiple of 9.

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

What is the value of A?

1) The four-digit number A77A is a multiple of 4.
2) The four-digit number A77A is a multiple of 9.

$$? = A\,\,\,\,\left( {\,A \in \left\{ {1,2,3, \ldots ,9} \right\}\,\,:\,\,{\rm{already}}\,\,{\rm{considering}}\,\,{\rm{each}}\,\,{\rm{statement}}\,\,{\rm{alone}}\,} \right)$$
$$\left( 1 \right)\,\,\,{{\left\langle {A77A} \right\rangle } \over 4} = {\mathop{\rm int}} \,\,\,\,\, \Leftrightarrow \,\,\,\,{{\left\langle {7A} \right\rangle } \over 4} = {\mathop{\rm int}} \,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,A = 2\,\,\,\,\,\, \hfill \cr
\,{\rm{Take}}\,\,A = 6\,\, \hfill \cr} \right.$$
$$\left( 2 \right)\,\,\,{{\left\langle {A77A} \right\rangle } \over 9} = {\mathop{\rm int}} \,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{{2 \cdot A + 14} \over 9} = {\mathop{\rm int}} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,A = 2\,\,\,\,\,\,\left( {\,9 \cdot k = 14 + 2 \cdot A\,\,,\,\,k > 2\,\,\,\, \Rightarrow \,\,\,\,A > 9} \right)$$

The correct answer is therefore (B).


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 (A) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

Condition 1)
If the last two digit is a multiple of 4, the original number is a multiple of 4.
Thus, A is 2 or 6 since 72 and 76 are multiples of 4.
Condition 1) is not sufficient, because we don't have a unique answer

Condition 2)
If the sum of all digits is a multiple of 9, the original number is a multiple of 9.
A + 7 + 7 + A = 2A + 14
If 2A + 14 = 18, we have 2A = 4 or A = 2.
We don't have any digit integer A such that 2A = 9, 27 or 36.
Thus A = 2 is the unique integer.
Condition 2) is sufficient.

Answer: C

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.