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Algebra

by kaarthikvs » Wed Apr 17, 2013 1:37 am
DS1) Is x and integer ?
1)x^3 is an integer 2)3x is an integer

DS2)If x and y are 2 distinct integers such that xy+x+y+1=0, what is the value of x?
1)x=2n, n is an integer 2)x> y

Please help me with these questions.
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by Anju@Gurome » Wed Apr 17, 2013 1:41 am
kaarthikvs wrote:Is x and integer ?

1) x³ is an integer
2) 3x is an integer
Statement 1: As x³ is an integer, x need not be an integer.
Consider the following two scenarios,
  • x = 1 ---> x and x³ both are integers
    x = cube root of 2 ---> x is not an integer but x³ is
Not sufficient

Statement 2: If 3x is an integer, x need not be an integer.
Consider the following two scenarios,
  • x = 1 ---> x and 3x both are integers
    x = 1/3 ---> x is not an integer but 3x is
Not sufficient

1 and 2 Together: Let us assume 3x = I, where I is some integer.
So, x = I/3 ---> x³ = I³/27
As x³ is an integer and I is also an integer, I³ must be divisible by 27.
Hence, I is divisible by 3.
So, x is an integer.

The correct answer is C.
Last edited by Anju@Gurome on Wed Apr 17, 2013 2:11 am, edited 1 time in total.
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by Anju@Gurome » Wed Apr 17, 2013 1:47 am
kaarthikvs wrote:If x and y are 2 distinct integers such that xy + x + y + 1 = 0, what is the value of x?

1)x = 2n, n is an integer
2)x > y
--> xy + x + y + 1 = 0
--> x(y + 1) + (y + 1) = 0
--> (x + 1)(y + 1) = 0

So, either (x + 1) = 0 OR (y + 1) = 0 OR both (x + 1) and (y + 1) = 0
--> either x = -1 OR y = -1 OR x = y = -1

As x and y are two different integers, the third scenario is not possible here.
So, either x = -1 OR y = -1

Now, consider the following two cases,
  • x = 2 and y = -1
    x = 4 and y = -1
Both of the above examples satisfy both the statements but value of x is different.
hence, both statements together are not sufficient.

The correct answer is E.
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by kaarthikvs » Wed Apr 17, 2013 2:06 am
Thanks very much Anju for your swift reply. I have one more clarification regarding DS1) if the statement 1 is x^2 instead of x^3 then the answer will be E. correct ?
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by Anju@Gurome » Wed Apr 17, 2013 2:17 am
kaarthikvs wrote:Thanks very much Anju for your swift reply. I have one more clarification regarding DS1) if the statement 1 is x^2 instead of x^3 then the answer will be E. correct ?
No. The answer will be C as explained.
My initial post had a mistake. I have corrected that.
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