by: gmataccount123
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Difficulty:
75%
(hard)
Question Stats:
53% (02:25) correct
47%
(02:30)
wrong
based on 207
sessions
History
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Not Attempted Yet
If a and b are integers then what is the remainder when \(a^2+b^2\) is divided by 7?
(1) \(a - b = 4\)
(2) \(a^2b^3\) is divisible by 14
(1) \(a - b = 4\)
(2) \(a^2b^3\) is divisible by 14
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:
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Data Sufficiency Forum
ArunSharma12
Joined: 24 Oct 2015
Last visit: 20 Jul 2022
Posts: 512
Location: India
Schools: IIML-IPMX '22 (A)
GMAT 1: 650 Q48 V31
GMAT 2: 720 Q49 V38 (Online)
GPA: 4
Products:
Kritisood
statement 1:
\((a - b)^2 = a^2+b^2-2ab = 16\)
\(a^2+b^2 = 16 - 2ab\)
we do not know values of a,b.
not sufficient
statement 2:
\(a^2b^3 = 2*7*k\)
either a,b or a*b is a multiple of 14
not sufficient
combining both statements,
\(a^2+b^2 = \frac{rem(16 - 2ab)}{14}\) = 2
Statement 1 and 2 don't provide any value of a and b separately hence, insufficient.
Combined:
a^2+b^2 = (a-b)^2 + 2ab
We need to find the remainder of \(\frac{(a-b){^2}}{7}\) and \(\frac{2ab}{7}\)
From statement 1 we know the value of a-b and hence can find the remainder of part a of the equation
From statement 2 we know that a^2b^3 has a 7 as a prime factor, therefore, ab will also have 7 as a prime factor, thus the remainder of 2ab is 0
C is sufficient
Combined:
a^2+b^2 = (a-b)^2 + 2ab
We need to find the remainder of \(\frac{(a-b){^2}}{7}\) and \(\frac{2ab}{7}\)
From statement 1 we know the value of a-b and hence can find the remainder of part a of the equation
From statement 2 we know that a^2b^3 has a 7 as a prime factor, therefore, ab will also have 7 as a prime factor, thus the remainder of 2ab is 0
C is sufficient
ArunSharma12
\(a^2+b^2 = 16 + 2ab \)
\(not 16 - 2ab \)
Kritisood
(1) \(a - b = 4\)
We are asked about \(a^2+b^2\), so can we convert a-b in this form.
\((a-b)^2=4^2........a^2+b^2-2ab=16........a^2+b^2=16+2ab\)
So we are looking for the remainder when 16+2ab is divided by 7.
But we do not know what is ab.
Insufficient
(2) \(a^2b^3\) is divisible by 14
As a and b are integers, at least one of a or b is surely divisible by 7, that is ab=7x
But we cannot work further for the value of a^2+b^2.
Insufficient
Combined
16+2ab=16+7x=2+14+7x will give a remainder 2 when the term is divided by 7.
Sufficient
C
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