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13 Jun 2010, 10:06
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62%
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Is the units digit of integer \(x^2 - y^2\) a zero?
(1) \(x - y\) is an integer divisible by 30
(2) \(x + y\) is an integer divisible by 70
(1) \(x - y\) is an integer divisible by 30
(2) \(x + y\) is an integer divisible by 70
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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
13 Jun 2010, 12:22
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Is the units digit of integer \(x^2 - y^2\) a zero?
Given: \(x^2-y^2=(x-y)(x+y)=integer\).
Question: is the units digit of this integer zero, which can be translated as is \(x^2-y^2\) divisible by 10.
(1) \(x - y\) is an integer divisible by 30. So, \(x-y=30m\) (where \(m\) is an integer). If \(x\) and \(y\) are integers, then \(x+y=integer\) and \(x^2-y^2=30m*(x+y)=30m*integer\), which is divisible by 10 BUT if \(x=30.75\) and \(y=0.75\), then \(x^2-y^2=(x-y)(x+y)=30*31.5=945=integer\), which is not divisible by 10. Not sufficient.
(2) \(x + y\) is an integer divisible by 70. So, \(x+y=70n\) (where \(n\) is an integer). If \(x\) and \(y\) are integers, then \(x-y=integer\) and \(x^2-y^2=(x-y)*70n=integer*70n\), which is divisible by 10 BUT if \(x=69.75\) and \(y=0.25\), then \(x^2-y^2=(x-y)(x+y)=69,5*70=4865=integer\), which is not divisible by 10. Not sufficient.
(1)+(2) \(x^2-y^2=(x-y)(x+y)=30m*70n=integer\), which is divisible by 10. Sufficient.
Answer: C
Hope it's clear.
Given: \(x^2-y^2=(x-y)(x+y)=integer\).
Question: is the units digit of this integer zero, which can be translated as is \(x^2-y^2\) divisible by 10.
(1) \(x - y\) is an integer divisible by 30. So, \(x-y=30m\) (where \(m\) is an integer). If \(x\) and \(y\) are integers, then \(x+y=integer\) and \(x^2-y^2=30m*(x+y)=30m*integer\), which is divisible by 10 BUT if \(x=30.75\) and \(y=0.75\), then \(x^2-y^2=(x-y)(x+y)=30*31.5=945=integer\), which is not divisible by 10. Not sufficient.
(2) \(x + y\) is an integer divisible by 70. So, \(x+y=70n\) (where \(n\) is an integer). If \(x\) and \(y\) are integers, then \(x-y=integer\) and \(x^2-y^2=(x-y)*70n=integer*70n\), which is divisible by 10 BUT if \(x=69.75\) and \(y=0.25\), then \(x^2-y^2=(x-y)(x+y)=69,5*70=4865=integer\), which is not divisible by 10. Not sufficient.
(1)+(2) \(x^2-y^2=(x-y)(x+y)=30m*70n=integer\), which is divisible by 10. Sufficient.
Answer: C
Hope it's clear.
22 Nov 2011, 15:18
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Hello Bunuel,
Thanks very much for your explanation to this question, which by the way I came across while working through your excellent "700+ GMAT Data Sufficiency Questions With Explanations" document (wow -- they're tough!)
I had a quick question on this particular question. Firstly, I believe it is the case that x^n - y^n is always divisible by (x-y), and is divisible by (x+y) when the powers of x and y are even.
If I have not mis-stated, then would you mind pointing out the flaw in the following reasoning?
(1) "x-y is an integer divisible by 30" >> adding in the rule that "if a is a factor of b and b is a factor of c, then a is a factor of c", then: x^n - y^n is an integer divisible by (x-y), and (x-y) is an integer divisible by 30, then does it not follow that x^n - y^n is also an integer divisible by 30? and this is sufficient to say that x^n - y^n must end in zero.
(2) similar logic applies since the power of x and y are indeed even and therefore divisible by (x+y).
Would be very grateful if you could take 2 mins to point out the error in that alternative method.
Thanks very much for your contributions and for taking the time to help.
Thanks very much for your explanation to this question, which by the way I came across while working through your excellent "700+ GMAT Data Sufficiency Questions With Explanations" document (wow -- they're tough!)
I had a quick question on this particular question. Firstly, I believe it is the case that x^n - y^n is always divisible by (x-y), and is divisible by (x+y) when the powers of x and y are even.
If I have not mis-stated, then would you mind pointing out the flaw in the following reasoning?
(1) "x-y is an integer divisible by 30" >> adding in the rule that "if a is a factor of b and b is a factor of c, then a is a factor of c", then: x^n - y^n is an integer divisible by (x-y), and (x-y) is an integer divisible by 30, then does it not follow that x^n - y^n is also an integer divisible by 30? and this is sufficient to say that x^n - y^n must end in zero.
(2) similar logic applies since the power of x and y are indeed even and therefore divisible by (x+y).
Would be very grateful if you could take 2 mins to point out the error in that alternative method.
Thanks very much for your contributions and for taking the time to help.
, or have got it wrong. Could someone please explain? 
