Events & Promotions
|
|
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.
Jun 2207:30 PM EDT
-09:30 PM EDT
Master the GMAT with expert live instruction, a personalized study plan, and real-time support. Includes 40 hours of online classes plus 6 months of access to the TTP GMAT OnDemand video course. Class date: Mon/Wed June 22, 2026 →August 26, 2026
12 Mar 2023, 10:09
Kudos
Bookmarks
B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
(hard)
Question Stats:
57% (02:23) correct
43%
(02:36)
wrong
based on 115
sessions
History
Date
Time
Result
Not Attempted Yet
If x is a positive integer, is x a multiple of 3?
(1) x^3 - x is a multiple of 3.
(2) k^3 - 2k^2 + k is not a multiple of 3; x=k-2.
(1) x^3 - x is a multiple of 3.
(2) k^3 - 2k^2 + k is not a multiple of 3; x=k-2.
12 Mar 2023, 12:46
Kudos
Bookmarks
TBT
Statement 1
(1) x^3 - x is a multiple of 3.
\(x(x^2 - 1)\)
x(x+1)(x-1) is a multiple of three
It is given that x is an integer, hence x(x+1)(x-1) is a product of three consecutive integers.
In this expression any of the integers can be a multiple of 3, it can be either x or (x-1) or (x+1).
Hence the statement is not sufficient.
We can eliminate A and D.
Statement 2
(2) k^3 - 2k^2 + k is not a multiple of 3; x=k-2.
\(k(k^2 - 2k +1)\)
\(k(k-1)^2\)
\(k = x + 2\)
\((x+2)(x+1)^2\) is not a multiple of 3
As x and k are integers, we can conclude that (x+2)(x+1) is not a multiple of.
A set of three consecutive integers is always a multiple of 3. As (x+2) and (x+1) is not a multiple, we can conclude that x is a multiple of 3.
The statement is sufficient.
Option B
12 Mar 2023, 23:58
Kudos
Bookmarks
TBT
Solution:
Pre Analysis:
- x is a positive integer
- We are asked if x a multiple of 3 or not
- This is a YES-NO question
Statement 1: x^3 - x is a multiple of 3
- \(x^3-x\)
\(⇒x(x^2-1)\)
\(⇒(x-1)x(x+1)\) is a multiple of 3 - We already know that \((x-1)x(x+1)\) will be a multiple of 6 if x is a positive integer
- So, the information in this statement is an obvious fact
- Thus, statement 1 alone is not sufficient and we can eliminate options A and D
Statement 2: k^3 - 2k^2 + k is not a multiple of 3; x=k-2
- \(k^3 - 2k^2 + k\)
\(⇒(x+2)^3-2(x+2)^2+(x+2)\)
\(⇒(x+2)[(x+2)^2-2(x+2)+1]\)
\(⇒(x+2)(x^2+4+4x-2x-4+1)\)
\(⇒(x+2)(x^2+2x+1)\)
\(⇒(x+2)(x+1)^2\)
\(⇒(x+1)(x+1)(x+2)\)
\(⇒x(x+1)(x+2)+(x+1)(x+2)\) - We know that \(x(x+1)(x+1)\) is always divisible by 6 and 3
- So, if \(x(x+1)(x+2)+(x+1)(x+2)\) is not divisible by 3, then (x+1)(x+2) is not divisible by 3
- This is only possible when x is a multiple of 3
- Thus, statement 2 alone is sufficient
Hence the right answer is Option B
