Data Sufficiency

[GMAT math practice question]

a and b are positive integers. What is the value of 2^a + 2^b?

1) a is the units digit of 7^1020 and b is the units digit of 3^224.
2) a and b are neither prime numbers nor composite numbers.

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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.
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Since we have 2 variables (a and b) and 0 equations and each condition has 2 equations, C is most likely to be the answer. Let’s look at both conditions together. However, since the value of condition (1) is equal to the value of condition (2), by Tip 1, we get D as the most likely answer. Let’s look at each condition separately.

Let’s look at the condition 1). It tells us that a = 1 and b = 1.

Units of powers of 7 are 7^1~7, 7^2~9, 7^3~3, 7^4~1, 7^5~7, …
So, the units digits of 7^n have a period of 4:
They form the cycle 7 -> 9 -> 3 -> 1.
Thus, 7^n has a units digit of 1 if n has a remainder of 0 when it is divided by 4.
The remainder is 0 when 224 is divided by 4, so the units digit of 7^1020 is 1.

Units of powers of 3 are 3^1~3, 3^2~9, 3^3~7, 3^4~1, 3^5~3, …
So, the units digits of 3^n have a period of 4:
They form the cycle 3 -> 9 -> 7 -> 1.
Thus, 3^n has a units digit of 1 if n has a remainder of 0 when it is divided by 4.
The remainder is 0 when 224 is divided by 4, so the units digit of 3224 is 1.

2^1 + 2^1 = 2 + 2 = 4.

The answer is unique, so the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Let’s look at the condition 2), it tells us that a = 1 and b = 1.

2^1 + 2^1 = 2 + 2 = 4.

The answer is unique, so the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Also, the original condition needs 2 equations.
Condition 1) has 2 equations.
Condition 2) has 2 equations.

Each condition ALONE is sufficient.

Therefore, D is the correct answer.
Answer: D

Note: Tip 1) of the VA method states that D is most likely the answer if condition 1) gives the same information as condition 2).

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.