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GMAT 1: 720 Q49 V40
Posts: 939
Updated on: 30 Jul 2019, 03:26
Originally posted by freedom128 on 30 Jul 2019, 00:00.
Last edited by freedom128 on 30 Jul 2019, 03:26, edited 1 time in total.
Last edited by freedom128 on 30 Jul 2019, 03:26, edited 1 time in total.
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
51% (01:59) correct
49%
(02:17)
wrong
based on 139
sessions
History
Date
Time
Result
Not Attempted Yet
If \(a\) and \(b\) are positive integers, what is the remainder of \(a^3-b^3\) when divided by 3?
(1) When \((a-b)^2\) is divided by 3, the remainder is 0
(2) When \(a^2+b^2\) is divided by 3, the remainder is 2
(1) When \((a-b)^2\) is divided by 3, the remainder is 0
(2) When \(a^2+b^2\) is divided by 3, the remainder is 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
Statement 1
\((a-b)^2= 0 mod 3\), and a and b are integers
Hence a-b=0 mod 3
\(a^3-b^3=(a-b)(a^2+b^2+ab)\)
As a-b is a multiple of 3, \(a^3-b^3\) must be a multiple of 3
Sufficient
Statement 2-
\(a^2+b^2\)= 2 mod 3
There are 3 cases possible
1) a=1 mod 3, and b=1 mod 3
ab=1 mod 3
\(a^2+b^2+ab\)=0 mod 3
Hence \(a^3-b^3\)=0 mod 3
2) a=2 mod 3 and b= 1 mod 3
a-b=1 mod 3
and \(a^2+b^2+ab\)=2 mod 3
\(a^3-b^3\)=2 Mod 3
3) a=2 mod 3, and b=2 mod 3
a-b=0 mod 3
\(a^3-b^3\)=0 mod 3
Insufficient
\((a-b)^2= 0 mod 3\), and a and b are integers
Hence a-b=0 mod 3
\(a^3-b^3=(a-b)(a^2+b^2+ab)\)
As a-b is a multiple of 3, \(a^3-b^3\) must be a multiple of 3
Sufficient
Statement 2-
\(a^2+b^2\)= 2 mod 3
There are 3 cases possible
1) a=1 mod 3, and b=1 mod 3
ab=1 mod 3
\(a^2+b^2+ab\)=0 mod 3
Hence \(a^3-b^3\)=0 mod 3
2) a=2 mod 3 and b= 1 mod 3
a-b=1 mod 3
and \(a^2+b^2+ab\)=2 mod 3
\(a^3-b^3\)=2 Mod 3
3) a=2 mod 3, and b=2 mod 3
a-b=0 mod 3
\(a^3-b^3\)=0 mod 3
Insufficient
chondro48
GMAT 1: 720 Q49 V40
Posts: 939
nick1816
Excellent explanation. Well done!
Posted from my mobile device
Posts: 28
1. Statement 1 says that [(a-b)(a-b)]/3 leaves no reminder. So (a-b) must be divisible by 3.
2. Statement 2 says [(a+b)^2- 2ab]= 2, means 3-[(a+b)^2]/3+[3-(2ab/3)]= 2. It could be 1+1 or 2+0 or 0+2. Hence insufficient.
Hency answer= A
[size=80][b][i]Posted from my mobile device[/i][/b][/size]
2. Statement 2 says [(a+b)^2- 2ab]= 2, means 3-[(a+b)^2]/3+[3-(2ab/3)]= 2. It could be 1+1 or 2+0 or 0+2. Hence insufficient.
Hency answer= A
[size=80][b][i]Posted from my mobile device[/i][/b][/size]
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