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12 Mar 2023, 10:09
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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% (02:24) correct
43%
(02:36)
wrong
based on 116
sessions
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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.
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
12 Mar 2023, 12:46
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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
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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
