If x is an Integer, How Many Even Numbers Does Set GMAT Data Sufficiency

Question: If x is an integer, how many even numbers does list {0, x,\(x^2\),\(x^3\),…,\(x^9\)} contain?

1. The mean of the list is even
2. The standard deviation of the list is 0

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.

Correct Answer: D

Approach Solution (1):

We have the list with 10 terms: {0, x,\(x^2\), \(x^3\) ,…,\(x^9\)}.

Note that if x = odd then the list will contain one even (0) and 9 odd terms (as if x = odd, then \(x^2\)= odd, \(x^3\)= odd,…, \(x^9\)= odd) and if x = even then the list will contain all even terms (as if x = even, then\(x^2\)= even,\(x^3\) = even,…,\(x^9\) = even)

Also note that, the standard deviation is always more than or equal to zero: . SD is 0 only when the list contains all the identical elements (or which is same only 1 element).

(1) The mean of the list is even. Since mean = \(\frac{sum}{10}\)= even, then sum = 10 * even = even.

So, we have that 0 + x +\(x^2\)+\(x^3\)+…+\(x^9\)= even or x +\(x^2\)+\(x^3\)+…+\(x^9\)= even, which implies that x = even (if x = odd then the sum of 9 odd numbers would be odd). x = even means that all 10 terms in the list are even.
Sufficient.

(2) The standard deviation of the list is 0. According to the above, all 10 terms of the list are identical and since the first terms is 0, then all other terms must be equal to zero, hence all 10 terms in the list are even.
Sufficient.

Correct option: D

“If x is an integer, how many even numbers does list {0, x,\(x^2\),\(x^3\),…,\(x^9\)} contain?”- 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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