by: Reon
17 mins
|
|
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
Track
Your Progress
Practice
Pays
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
85%
(hard)
Question Stats:
43% (01:37) correct
57%
(01:14)
wrong
based on 425
sessions
History
Date
Time
Result
Not Attempted Yet
Is x an even integer?
(1) x^2 is divisible by 4.
(2) x^4 is divisible by 16.
Kudos for a correct solution.
_________________
(1) x^2 is divisible by 4.
(2) x^4 is divisible by 16.
Kudos for a correct solution.
Pure algebraic questions like this one are no longer part of the GMAT Data Sufficiency syllabus.
For relevant Data Sufficiency questions, please check the Data Sufficiency forum:
Data Sufficiency Forum
Data Sufficiency Forum
_________________
New to the GMAT Club?
- • Questions: Question Bank | Bunuel's Signature Collection
• Tests: GMAT Club Tests | Forum Quiz
• Guides: Quantitative | Critical Reasoning | Reading Comprehension | Ultimate Quant Guide | All You Need for Quant
• Rules: Quantitative | Verbal
Do St(1) and St(2) imply x^2/4 and x^4/16 have integer solutions? Since this is 700 level I'm just getting caught up on the wording of the problem.
Either way, if St(1) and St(2) aren't constrained by integer solutions the answer is E. The number 1 can be used to disprove each solution.
Additionally the square root of 8 squared is divisible by 4 in St(1) which makes X a non-integer. The square root of 8^4 is also divisible by 16, however x is still a non integer.
Answer choice - E.
Either way, if St(1) and St(2) aren't constrained by integer solutions the answer is E. The number 1 can be used to disprove each solution.
Additionally the square root of 8 squared is divisible by 4 in St(1) which makes X a non-integer. The square root of 8^4 is also divisible by 16, however x is still a non integer.
Answer choice - E.
ak1802
x^2 is divisible by 4 means:
1. x^2 is an integer;
2. x^2/4 is an integer.
Generally, on the GMAT when we are told that \(a\) is divisible by \(b\) (or which is the same: "\(a\) is a multiple of \(b\)", or "\(b\) is a factor of \(a\)"), we can say that:
(i) \(a\) is an integer;
(ii) \(b\) is an integer;
(iii) \(\frac{a}{b}=integer\).
_________________
New to the GMAT Club?
- • Questions: Question Bank | Bunuel's Signature Collection
• Tests: GMAT Club Tests | Forum Quiz
• Guides: Quantitative | Critical Reasoning | Reading Comprehension | Ultimate Quant Guide | All You Need for Quant
• Rules: Quantitative | Verbal
My thought for answering the question was:
If X^2 is divisible by 4, we can think:
X^2 = 4a, where a is a number from 1 to inf, so
sqrt(x^2) = x = 2 * sqrt(a).
Since a can be any number, sqrt(a) is not constrained to an integer. The same logic can be applied to the second statement; therefore, the answer is (E)
Cheers
If X^2 is divisible by 4, we can think:
X^2 = 4a, where a is a number from 1 to inf, so
sqrt(x^2) = x = 2 * sqrt(a).
Since a can be any number, sqrt(a) is not constrained to an integer. The same logic can be applied to the second statement; therefore, the answer is (E)
Cheers
Bunuel
Thanks for the clarification. Whenever I see a question categorized as 700+ l am much more skeptical, sometimes forgetting how the verbiage is used.
Either way, that means x^2/4 and x^4/8 must be integers. This allows for integer solutions for X such as 2,4,8, etc....however it also allows for non-integer solutions for X such as the square root of 8. (sqrt8)^2/4 = integer. (sqrt8)^4/4 = integer.
Therefore answer is E.
Harley1980
Retired Moderator
Joined: 06 Jul 2014
Last visit: 14 Jun 2024
Posts: 995
Location: Ukraine
Concentration: Entrepreneurship, Technology
GMAT 1: 660 Q48 V33
GMAT 2: 740 Q50 V40
GMAT 2: 740 Q50 V40
Posts: 995
Bunuel
1) At least two possible variants: \(x = 2\) or \(x = \sqrt{12}\)
Insufficient
2) At least two possible variants: \(x = 2\) or \(x = ^4\sqrt{32}\)
Insufficient
1+2) still Insufficient: \(x = 2\) or \(x = ^4\sqrt{32}\)
Answer is E
_________________
bsattin1
your logic is incorrect.
In the form of x^2 /4 = a ; a must be an integer, because x^2 is divisible by 4. This results in an integer solution. Therefore a is always an integer.
X, however is not always constrained to an integer. sqrt(8)^2 / 4 = 2. Furthermore sqrt(8)^4/16 = 4.
The logic on quant problems is critical. If you can understand the basis for most of these questions, and their constraints, they're much easier to solve.
Bunuel
MANHATTAN GMAT OFFICIAL SOLUTION:
The problem is asking whether x is even – that is, whether x is divisible by 2.
Statement (1): NOT SUFFICIENT. It’s easy to look for “confirming” evidence by assuming the question. That’s a bad strategy. Rather, take the statement as true, and see whether you answer the question the same way every time. In particular, with a Yes/No question, be a skeptic. Try to find both a Yes case and a No case.
The statement says that x^2 is divisible by 4. That is, x^2 is a multiple of 4. List a few multiples of 4 and then answer the question using those cases.
First case: x^2 = 4 itself. So then x = 2 or –2. In both scenarios, x is indeed even, so the answer is Yes.
Second case: x^2 = 8. So then x = the positive or negative square root of 8 (√8 or –√8). Since 8 is not a perfect square, x is not an integer (the positive root is between 2 and 3). Only integers can be even, so the answer to the question is No.
Since we have a Yes case and a No case, we can stop. This statement is insufficient.
Statement (2): NOT SUFFICIENT. As in the previous analysis, you should take the statement as true, then apply different cases to the question and see whether you get the same answer each time.
The statement says that x^4 is divisible by 16. That is, x^4 is a multiple of 16. List a few multiples of 16 and then answer the question using those cases.
First case: x^4 = 16 itself. So then x = 2. This is an even integer, so the answer is Yes.
Second case: x^4 = 32. So then x = the fourth root of 32. Since 32 is not a perfect fourth power, x is not an integer, let alone an even integer. So the answer is No.
Statements (1) and (2) together: NOT SUFFICIENT. There are still Yes and No cases. In fact, the first statement implies the second statement.
If x^2 is divisible by 4, then we can write x^2 = 4n, where n is an integer.
Squaring both sides gives us x^4 = 16n^2, and since n^2 is automatically an integer if n is an integer, the second statement is also satisfied.
So the second statement adds nothing to the first statement, which is more restrictive. Every case that satisfies the first statement satisfies the second statement as well. Together, the statements are NOT sufficient.
The correct answer is E.
_________________
New to the GMAT Club?
- • Questions: Question Bank | Bunuel's Signature Collection
• Tests: GMAT Club Tests | Forum Quiz
• Guides: Quantitative | Critical Reasoning | Reading Comprehension | Ultimate Quant Guide | All You Need for Quant
• Rules: Quantitative | Verbal
Bunuel
Hi Bunuel,
Sorry for jumping into the old thread, could you please explain more and give concrete example about the case for not choosing C?
Thank you
akara2500
x = 2 satisfies both statements and gives an YES answer to the question.
\(x = \sqrt{8}\) satisfies both statements and gives a NO answer to the question (\(\sqrt{8}\) is NOT an integer and hence cannot be even, recall that only integers can be even or odd).
Hope it helps.
_________________
New to the GMAT Club?
- • Questions: Question Bank | Bunuel's Signature Collection
• Tests: GMAT Club Tests | Forum Quiz
• Guides: Quantitative | Critical Reasoning | Reading Comprehension | Ultimate Quant Guide | All You Need for Quant
• Rules: Quantitative | Verbal
Posts: 7
[ltr]Each statement alone is sufficient to determine if x is an even integer.
Here's why:
Statement (1): x2 is divisible by 4.
If x is an integer:
Thus, statement (1) alone is sufficient.
[hr]
Statement (2): x4 is divisible by 16.
If x is an integer:
Thus, statement (2) alone is sufficient.
Since each statement alone is sufficient to determine that x is an even integer, the answer is that either statement alone is sufficient.[/ltr]
Here's why:
Statement (1): x2 is divisible by 4.
If x is an integer:
- If x is even, then x can be written as 2k for some integer k. So, x2=(2k)2=4k2. This is clearly divisible by 4.
- If x is odd, then x can be written as 2k+1 for some integer k. So, x2=(2k+1)2=4k2+4k+1=4(k2+k)+1. This leaves a remainder of 1 when divided by 4, so it's not divisible by 4.
Thus, statement (1) alone is sufficient.
[hr]
Statement (2): x4 is divisible by 16.
If x is an integer:
- If x is even, then x can be written as 2k for some integer k. So, x4=(2k)4=16k4. This is clearly divisible by 16.
- If x is odd, then x can be written as 2k+1 for some integer k. An odd number raised to any positive integer power is always odd. Therefore, x4 would be an odd number. An odd number cannot be divisible by 16 (an even number). For example, if x=1, x4=1, not divisible by 16. If x=3, x4=81, not divisible by 16.
Thus, statement (2) alone is sufficient.
Since each statement alone is sufficient to determine that x is an even integer, the answer is that either statement alone is sufficient.[/ltr]
Posts: 7
[ltr]Each statement alone is sufficient to determine if x is an even integer.
Here's why:
Statement (1): x2 is divisible by 4.
If x is an integer:
Thus, statement (1) alone is sufficient.
[hr]
Statement (2): x4 is divisible by 16.
If x is an integer:
Thus, statement (2) alone is sufficient.
Since each statement alone is sufficient to determine that x is an even integer, the answer is that either statement alone is sufficient.[/ltr]
Here's why:
Statement (1): x2 is divisible by 4.
If x is an integer:
- If x is even, then x can be written as 2k for some integer k. So, x2=(2k)2=4k2. This is clearly divisible by 4.
- If x is odd, then x can be written as 2k+1 for some integer k. So, x2=(2k+1)2=4k2+4k+1=4(k2+k)+1. This leaves a remainder of 1 when divided by 4, so it's not divisible by 4.
Thus, statement (1) alone is sufficient.
[hr]
Statement (2): x4 is divisible by 16.
If x is an integer:
- If x is even, then x can be written as 2k for some integer k. So, x4=(2k)4=16k4. This is clearly divisible by 16.
- If x is odd, then x can be written as 2k+1 for some integer k. An odd number raised to any positive integer power is always odd. Therefore, x4 would be an odd number. An odd number cannot be divisible by 16 (an even number). For example, if x=1, x4=1, not divisible by 16. If x=3, x4=81, not divisible by 16.
Thus, statement (2) alone is sufficient.
Since each statement alone is sufficient to determine that x is an even integer, the answer is that either statement alone is sufficient.[/ltr]
Moderator:
Announcements
How to Tell If Your GMAT Prep Resources Are Actually Helping You
Subscribe to GMAT Club for New Podcast Episodes, Debriefs, & Insightful Sessions on GMAT Preparation
Subscribe to GMAT Club for New Podcast Episodes, Debriefs, & Insightful Sessions on GMAT Preparation
Tuck at Dartmouth
AI at Dartmouth Tuck: How the School Is Preparing Leaders for the Age of Artificial Intelligence
Inside the courses, research, and hands-on learning defining wise, decisive leadership in an AI-driven world.
Read More
Inside the courses, research, and hands-on learning defining wise, decisive leadership in an AI-driven world.
Read More
Top 5 GMAT Debrief Videos
Success stories and strategies from high-scoring candidates.