Data Sufficiency

[GMAT math practice question]

If (-4)^3 \(\div\) (-2)^m=-2^(n-6) what is the value of mn?

1) m and n are positive even integers.
2) m < 6 and n > 6.

=>
Forget conventional ways of solving math questions. F or 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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question.
(-4)^3 ÷ (-2)^m=-2^(n-6)
=> -2^6 ÷ (-2)^m = -2^(n-6)
=> -2^(6-m) = -2^(n-6), where m is an even integer.
=> 6 - m = n - 6
=> m + n = 12.

Follow the second and the third step: From the original condition, we have 2 variables (m and n) and 1 equation (m + n = 12). To match the number of variables with the number of equations, we need 1 more equation. Since conditions (1) and (2) will provide 1 equation each, D would most likely be the answer.

Recall 3 Principles and choose D as the most likely answer. Let’s look at each condition separately,

Condition (1) tells us that m and n are positive even integers, from which we get (m, n) = (2, 10), (4, 8), (6, 6), (8, 4), and (10, 4). If m = 2, and n = 10 we get mn = 20 and if m = 4, and n = 8, we get mn = 32.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition (2) tells us that m < 6 and n > 6, from which we cannot get the unique values of m and n. For example, if m = 2, and n = 10, then we get mn = 20 and if m = 4, n = 8, we get mn = 32.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions (1) and (2) together also do not give us unique values for m and n. For example, if m = 2 and n = 10, we get mn = 20 and if m = 4, and n = 8, we get mn = 32.

The answer is not unique, so both conditions (1) and (2) combined are not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions (1) and (2) together are not sufficient.
Therefore, E is the correct 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.