Question: If x is a non-zero integer, what is the value of x?

1. \(x^{4x}=x^{16}\)
2. \((x^x) (x^2)=x^6\)

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

Answer:
Solution with Explanation:
Approach Solution (1):

(1) \(x^{4x}=x^{16}\)

Clearly, when base is 1, the power could be anything. Thus x = 1 is a possibility.
Equation the power as the base is x: 4x = 14 or x = 4
Insufficient

(2) \((x^x) (x^2)=x^6\)

Clearly, when base is 1, the power could be anything. Thus x = 1 is a possibility.
Also, \(x^{x+2}=x^6\). Equating the power as the base is x: x + 2 = 6 or x = 4
Insufficient
Combined
x can still be 1 or 4
Insufficient

Correct Option: E

Approach Solution (2):

(1) \(x^{4x}=x^{16}\)

Let’s begin by assuming \(x\neq0,x\neq1, and x\neq-1\)

Since the bases are equal, the exponents are equal, which means we can write: 4x = 16
So, x = 4 is one solution (so far)

We must now test the provisos (other than x = 0, since we are told x is a non-zero integer)

First let’s test x = 1 by plugging it into the equation to get: \(1^{4(1)}=1^{16}\)

Evaluate to get 1 = 1. WORKS!
So, x = 1 is another possible solution.
Now let’s test x = -1 by plugging it into the equation to get: \((-1)^{4(-1)}=(-1)^{16}\)

Evaluate to get 1 = 1. WORKS!
So, x = -1 is another possible solution
Since, x can equal 4, 1, or -1, statement 1 is NOT SUFFICIENT

(2) \((x^x) (x^2)=x^6\)

First apply the product law to get: \(x^{x+2}=x^6\)

Once again let’s assume \(x\neq0,x\neq1, and x\neq-1\)

Since the bases are equal, the exponents are equal, which means we can write: x + 2 = 6
So, x = 4 is one solution (so far)

Now let’s test x = 1 by plugging it into the equation to get: \(1^{1+2}=1^6\)

Evaluate to get 1 = 1. WORKS!
So, x = 1 is another possible solution.

Now, let’s test x = -1 by plugging it into the equation to get: \(1^{(-1)+2}=1^6\)

Evaluate to get -1 = 1. Doesn’t work
Since, x can equal 4 or 1, statement 2 is not sufficient
(1) + (2)

S1 tells us that x can equal 4, 1, or -1
S2 tells us that x can equal 4 or 1
So when we combine the statements, we know that x can equal 4 or 1

Since we can’t answer the target question with certainty, the combined statements are not sufficient.

Correct Option: E

“If x is a non-zero integer, what is the value of x?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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