Algebra

[kaarthikvs]

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.

[Anju@Gurome]

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.

[Anju@Gurome]

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.

[kaarthikvs]

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 ?

[Anju@Gurome]

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.