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# If x and y are integers, is x^3-y^3 an odd integer?

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 4707
GPA: 3.82
If x and y are integers, is x^3-y^3 an odd integer? [#permalink]

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15 Dec 2017, 07:24
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[GMAT math practice question]

If $$x$$ and $$y$$ are integers, is $$x^3-y^3$$ an odd integer?

1) $$x$$ is an odd number
2) $$x$$ and $$y$$ are consecutive integers
[Reveal] Spoiler: OA

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Joined: 02 Aug 2009
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Re: If x and y are integers, is x^3-y^3 an odd integer? [#permalink]

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16 Dec 2017, 01:29
MathRevolution wrote:
[GMAT math practice question]

If $$x$$ and $$y$$ are integers, is $$x^3-y^3$$ an odd integer?

1) $$x$$ is an odd number
2) $$x$$ and $$y$$ are consecutive integers

For $$x^3-y^3$$ to be an integer, both x and y should be different from each other, one should be ODD and other EVEN..

(1) X is odd..
Y is odd....NO
y is even....YES
Insufficient

(2). X and y are consecutive integers..
One will be ODD and other EVEN..
Ans is YES
Sufficient

B
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Re: If x and y are integers, is x^3-y^3 an odd integer? [#permalink]

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16 Dec 2017, 08:23
MathRevolution wrote:
[GMAT math practice question]

If $$x$$ and $$y$$ are integers, is $$x^3-y^3$$ an odd integer?

1) $$x$$ is an odd number
2) $$x$$ and $$y$$ are consecutive integers

An odd integer raised to any positive integral power will be odd, and similarly an even integer raised to any integral power will be even. So if x is odd, then x^3 will also be odd, and if x is even, x^3 will be even. Same goes for y. Also difference of two integers will be odd ONLY if one of them is odd and the other is even. Else their difference will be even.

(1) x is odd, but nothing given about y. So Insufficient.

(2) x and y are consecutive integers, that means one of them will be odd and the other even. So as explained above, one of x^3 and y^3 will be odd and the other even. So the difference will be odd. Sufficient.

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Re: If x and y are integers, is x^3-y^3 an odd integer? [#permalink]

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16 Dec 2017, 10:15
MathRevolution wrote:
[GMAT math practice question]

If $$x$$ and $$y$$ are integers, is $$x^3-y^3$$ an odd integer?

1) $$x$$ is an odd number
2) $$x$$ and $$y$$ are consecutive integers

Hi chetan2u & amanvermagmat,

From Stmnt B if we take x=0, y = 1... then x^3 - y^3 = -1... Is odd, even defined for negative numbers??

Also, whenever it is mentioned in question that two number are consecutive, should the two numbers be in the a-1, a / a, a+1 format or a-d, a / a, a+d format where d = 1,2,3,... etc.

Can you please clarify my above doubt.
DS Forum Moderator
Joined: 21 Aug 2013
Posts: 676
Location: India
Re: If x and y are integers, is x^3-y^3 an odd integer? [#permalink]

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16 Dec 2017, 10:25
1
KUDOS
rahul16singh28 wrote:
MathRevolution wrote:
[GMAT math practice question]

If $$x$$ and $$y$$ are integers, is $$x^3-y^3$$ an odd integer?

1) $$x$$ is an odd number
2) $$x$$ and $$y$$ are consecutive integers

Hi chetan2u & amanvermagmat,

From Stmnt B if we take x=0, y = 1... then x^3 - y^3 = -1... Is odd, even defined for negative numbers??

Also, whenever it is mentioned in question that two number are consecutive, should the two numbers be in the a-1, a / a, a+1 format or a-d, a / a, a+d format where d = 1,2,3,... etc.

Can you please clarify my above doubt.

Hi rahul16singh28

An odd integer is that integer which is not divisible by 2. So 5, 17, 1, -19, -3, etc are all odd integers.
An even integer is that which is divisible by 2. So 20, 4, 2, -2, 0, -54 etc are all even integers.
So yes, even negative integers can be classified as odd and even. (And 0 is treated as an even integer, because its divisible by 2. 0 is actually divisible by all non-zero numbers)

Consecutive integers are those which are immediate integers with a difference of '1'. So if two integers are consecutive, we can write them as a & a+1 Or a-1 & a. Eg, -1, 0 are consecutive. 10, 11, 12, 13, 14 are five consecutive integers.

a-d, a, a+d means these three numbers are in Arithmetic Progression (AP). This means difference between second and first number is same as the difference between third and second number (and this difference is called 'common difference'). These numbers are not necessarily consecutive.

Eg, 12, 15, 18 are in AP, with a common difference of '3', but they are NOT consecutive integers.
-7, -5, -3 are also in AP, with a common difference of '2', but they are NOT consecutive integers.
-1, 0, 1 are also in AP, with a common difference of '1', hence they ARE consecutive integers.
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 4707
GPA: 3.82
Re: If x and y are integers, is x^3-y^3 an odd integer? [#permalink]

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16 Dec 2017, 23:08
rahul16singh28 wrote:
MathRevolution wrote:
[GMAT math practice question]

If $$x$$ and $$y$$ are integers, is $$x^3-y^3$$ an odd integer?

1) $$x$$ is an odd number
2) $$x$$ and $$y$$ are consecutive integers

Hi chetan2u & amanvermagmat,

From Stmnt B if we take x=0, y = 1... then x^3 - y^3 = -1... Is odd, even defined for negative numbers??

Also, whenever it is mentioned in question that two number are consecutive, should the two numbers be in the a-1, a / a, a+1 format or a-d, a / a, a+d format where d = 1,2,3,... etc.

Can you please clarify my above doubt.

-1, -3, -5, ... are odd integers too.
And 0, -2, -4, -6, ... are even integers.

a - 1, a / a, a+1 are consecutive integers,
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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 4707
GPA: 3.82
Re: If x and y are integers, is x^3-y^3 an odd integer? [#permalink]

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17 Dec 2017, 23:21
=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question, and then recheck the question.

We can modify the original condition and question as follows.
There are two different ways in which $$x^3-y^3$$ can be odd:
$$x$$ is even and $$y$$ is odd.
$$x$$ is odd and $$y$$ is even.

Since condition 2) tells us that x and y are consecutive integers, one of them must be odd, and the other must be even. In both cases, the answer is ‘yes’.
Therefore, condition 2) is sufficient.

As condition 1) gives us no information about y, it is not sufficient.

Normally, in problems which require 2 or more additional equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

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Re: If x and y are integers, is x^3-y^3 an odd integer?   [#permalink] 17 Dec 2017, 23:21
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# If x and y are integers, is x^3-y^3 an odd integer?

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