Question: Is x the square of an integer?

1. When x is divided by 12, the remainder is 6
2. When x is divided by 14, the remainder is 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.

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

S1: 6, 18, 30, 42, 54, 68, 80, 92, 104, 116, 128, 140, 152, 164, 178, 190, 202…
Now the obvious observation is that the answer to the question is always no. The less obvious observation is that the prime factorization of every number here has an odd power of 2, so the answer will continue to ne NO
Sufficient
S2: 2, 16 already we have one No and one yes answer
Insufficient

Correct Option: A

Approach Solution (2):

(1) x = 12n + 6 No square of an integer
(2) x = 14n + 2
For n = 1, x = 14 + 2 = 16 is a square of 4
n = 7 x = 98 + 2 = 100 is a square of 1 0
n = 23 x = 23 * 14 + 2 = 322 + 2 = 324 is a square of 18

Others are possible

Correct Option: A

Approach Solution (3):

Statement 1: When x is divided by 12, the remainder is 6
given that:
x = 12q + 6
x = 6 (2q + 1)
x =2⋅3(2q + 1)
(2q+1) is an odd number, the power of 2 in x will be odd, thus ,x, cannot be a perfect square. Statement 1 is SUFFICIENT

Statement 2: when x is divided by 14, the remainder is 2
Given that x=14p+2. So, x could be 2, 16, 30, etc. So, it is not certain that x is a perfect square.

Hence, statement 2 is NOT SUFFICIENT

Correct Option: A

“Is x the square of 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.

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