If n is a positive integer, is 3^4+3^{n+4} divisible by 5?

[Max@Math Revolution]

[GMAT math practice question]

If n is a positive integer, is 3^4+3^{n+4} divisible by 5?

1) n is an even integer.
2) 3^8 +3^{n+8} is divisible by 5.

[GMATGuruNY]

[GMAT math practice question]

If n is a positive integer, is 3^4+3^{n+4} divisible by 5?

1) n is an even integer.
2) 3^8 +3^{n+8} is divisible by 5.

3� + 3^(n+4) = 3� + (3^n)(3�) = 3�(1 + 3^n) = 3�(3^n + 1).
The expression in blue will be divisible by 5 only if (3^n + 1) is a multiple of 5.
Question stem, rephrased:
Is (3^n + 1) a multiple of 5?

Statement 1:
Case 1: n=2, with the result that 3^n + 1 = 3² + 1 = 10.
In this case, the answer to the rephrased question stem is YES.
Case 2: n=4, with the result that 3^n + 1 = 3� + 1 = 82
In this case, the answer to the rephrased question stem is NO.
INSUFFICIENT.

Statement 2:
3� + 3^(n+8) = 3� + (3^n)(3�) = 3�(1 + 3^n) = 3�(3^n + 1).
For the expression in red to be divisible by 5, (3^n + 1) must be a multiple of 5.
Thus, the answer to the rephrased question stem is YES.
SUFFICIENT.

The correct answer is B.

[Max@Math Revolution]

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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 of the conditions on their own first.

Condition 1)
If n = 2, then 3^4 + 3^{n+4} = 3^4 + 3^6 = 3^4(1+3^2) = 81*10 = 810, and the answer is "yes".
If n = 0, then 3^4 + 3^{n+4} = 3^4 + 3^4 = 3^4(1+1) = 81*2 = 162, and the answer is "no".
Condition 1) is not sufficient.

Condition 2)
Now, 3^8 + 3^{n+8} = 3^8(1+3^n) is divisible by, but 3^8 is not divisible by 5. Since 5 is a prime number, 1+3^n must be divisible by 5.
Thus, 3^4 + 3^{n+4} = 3^4(1+3^n) is also divisible by 5.
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.

[Jeff@TargetTestPrep]

[GMAT math practice question]

If n is a positive integer, is 3^4+3^{n+4} divisible by 5?

1) n is an even integer.
2) 3^8 +3^{n+8} is divisible by 5.

Statement One Alone:

n is an even integer.

If n = 2, 3^4 + 3^6 = 3^4(1 + 3^2) = 3^4(10), which is divisible by 5.

If n = 4, 3^4 + 3^8 = 3^4(1 + 3^4) = 3^4(82), which is not divisible by 5.

Statement one alone is not sufficient.

Statement Two Alone:

3^8 + 3^{n+8} is divisible by 5.

3^8 + 3^{n+8} = 3^4(3^4 + 3^{n+4})

We are given that 3^8 + 3^{n+8} is divisible by 5. So 3^4(3^4 + 3^{n+4}) is divisible by 5. However, since 3^4 is not divisible by 5, 3^4 + 3^{n+4} must be divisible by 5. Statement two alone is sufficient.

Answer: B