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28 Feb 2017, 04:16
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
37% (02:25) correct
63%
(02:30)
wrong
based on 163
sessions
History
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Time
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Not Attempted Yet
Can the total number of integers that divide x be expressed in the form of \(2k + 1\), where k is a positive integer?
(1) \(√12x\) is an integer
(2) The product of \(√x\) and \(√y\) is an integer, where the total number of factors of \(\frac{y}{3}\) is odd.
Answer Choices:
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Thanks,
Saquib
Quant Expert
e-GMAT
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(1) \(√12x\) is an integer
(2) The product of \(√x\) and \(√y\) is an integer, where the total number of factors of \(\frac{y}{3}\) is odd.
Answer Choices:
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Thanks,
Saquib
Quant Expert
e-GMAT
Register for our Free Session on Number Properties (held every 3rd week) to solve exciting 700+ Level Questions in a classroom environment under the real-time guidance of our Experts
27 Mar 2017, 05:29
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Solution
Steps 1 & 2: Understand Question and Draw Inferences
- • We know that a number in the form of 2k + 1 leaves a remainder of 1 when divided by 2, (that is, a number that leaves a remainder of 1 when divided by 2) is odd.
• So, we are asked to find if the number of factors of x is odd.
To Find: Is the number of factors of x odd?
- • We know that if a number has odd number of factors, it has to be a perfect square. So, we are (indirectly) asked to find if x is a perfect square?
Step 3: Analyze Statement 1 independently
- 1. √12x is an integer
\(\sqrt{12\mathrm x}=\;\sqrt{2^2\ast3\mathrm x}=\;2\surd3\mathrm x\) is an integer. For √3x to be an integer, x should contain an odd power of 3.
Now, if 3 occurs odd number of times in x, x can’t be a perfect square.
Hence statement-1 is sufficient to answer the question.
Step 4: Analyze Statement 2 independently
- 2. The product of √x and √y is an integer, where the total number of factors of y/3 is odd.
Statement-2 tells us that the total number of factors of \(\frac{\mathrm y}3\) is odd i.e. \(\frac{\mathrm y}3\) is a perfect square. Let’s assume \(\frac{\mathrm y}3=\mathrm z^2\), where z is an integer.
\(y = 3z^2\). Since \(z^2\) is always non-negative, y will also be a non-negative integer
So, we know that \(\surd\mathrm x\operatorname{ *}\;\surd3\mathrm z^2\) is an integer i.e. z√3x is an integer.
For √3x to be an integer, x should contain an odd power of 3. If 3 occurs odd number of times in x, x can’t be a perfect square
Hence statement-2 is sufficient to answer the question.
Step 5: Analyze Both Statements Together (if needed)
Since we have a unique answer from steps 3 and 4, this step is not required.
Answer: D
Thanks,
Saquib
Quant Expert
e-GMAT
Register for our Free Session on Number Properties (held every 3rd week) to solve exciting 700+ Level Questions in a classroom environment under the real-time guidance of our Experts
General Discussion
28 Feb 2017, 17:55
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D.
Question asked if the x has odd number total of factors.
S1: Plug in x = 3 & x = 27 --> both get √12x integer and both 3, 27 have even number total of factors. Sufficient
S2: y/3 has odd number total of factors --> y = 3A (with A is prime factorize of Y/3) --> y has even number total of factors.
In order for √xy integer, x must have odd exponent --> x has even number total of factors. Sufficient
Question asked if the x has odd number total of factors.
S1: Plug in x = 3 & x = 27 --> both get √12x integer and both 3, 27 have even number total of factors. Sufficient
S2: y/3 has odd number total of factors --> y = 3A (with A is prime factorize of Y/3) --> y has even number total of factors.
In order for √xy integer, x must have odd exponent --> x has even number total of factors. Sufficient
