Question: Is \(\frac{7^7}{7^x}\)  an integer?

1. \(0\leq{x}\leq7\)​
2. \(|x|=x^2\)

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.

“Is  \(\frac{7^7}{7^x}\) an integer?” – 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.

Answer:

Approach Solution 1

\(\frac{7^7}{7^x}=7^{7-x}\), so as long as x is an integer and x \(\leq\)7, the expression is an integer.

Now considering S(1), we have  \(0\leq{x}\leq7\)

If x is an integer from this range, then \(\frac{7^7}{7^x}\) will be an integer but if x is a fraction from this range, then \(\frac{7^7}{7^x}\) will NOT be an integer.

Now we will see the S(2), we have \(|x|=x^2\)

Do squaring on both sides, we will get:

\(x^2=x^4\)

\(x^2(x^2-1)=0\)

\(x^2(x-1)(x+1)=0\)

X = 0, 1, -1

For each of these value \(\frac{7^7}{7^x}\) IS an integer. So, we have an YES answer.

Correct option: B

Approach Solution 2

From \(|x|=x^2\) , it does not follow |x| = x

• If x is negative, we will have:

\(-x=x^2\) (Notice here that –x there is NOT negative, its negative positive)

x (x+1) = 0

x = 0 or x = -1

x = -1

• If x is positive or o, then we’d get:

\(x=x^2\)

x (x-1) = 0

x = 0 or x = 1

So, x = -1, 0, 1

Correct option: B

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