If x, y are integers, is x^2+x+y an odd integer?

[Max@Math Revolution]

[Math Revolution GMAT math practice question]

If x, y are integers, is x^2+x+y an odd integer?

1) x is an odd integer
2) y is an odd integer

[fskilnik@GMATH]

[Math Revolution GMAT math practice question]

If x, y are integers, is x^2+x+y an odd integer?

1) x is an odd integer
2) y is an odd integer

$$x,y\,\,{\rm{ints}}\,\,\,\left( * \right)$$
$${{\text{x}}^{\text{2}}} + x + y\,\, = \,\,\underbrace {x\left( {x + 1} \right)}_{\left( * \right)\,\,{\text{even}}} + y\,\,\mathop = \limits^? \,\,\,{\text{odd}}\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\boxed{\,\,y\,\,\mathop = \limits^? \,\,\,{\text{odd}}\,\,}$$
$$\left( 1 \right)\,\,x\,\,{\rm{odd}}\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,0} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.$$
$$\left( 2 \right)\,\,y\,\,{\rm{odd}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

[Brent@GMATPrepNow]

If x, y are integers, is x² + x + y an odd integer?

1) x is an odd integer
2) y is an odd integer

Given: x and y are integers

Target question: Is x² + x + y an odd integer?
This is a good candidate for rephrasing the target question.

Take x² + x + y and factor the first two terms to get: x(x + 1) + y
Notice that x and x+1 are CONSECUTIVE INTEGERS, which means one value is ODD and the other value must be EVEN.
As such, x(x + 1) will be EVEN for all values of x.
So, we can replace x(x + 1) with EVEN to get a new target question "Is EVEN + y an odd integer?"
At this point, we can see that, in order for EVEN + y to be odd, y must be ODD. So, we can REPHRASE the target question as....
REPHRASED target question: Is y ODD?

Aside: Here's a video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

Statement 1: x is an odd integer
This information will not help us answer the REPHRASED target question with certainty.
Statement 1 is NOT SUFFICIENT

Statement 2: y is an odd integer
Perfect!
The answer to the REPHRASED target question is YES, y is odd
Since we can answer the REPHRASED target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

[Max@Math Revolution]

=>

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. We then recheck the question.

The parity of x^2+x+y = x(x+1) + y is same as the parity of y, since x^2+x = x(x+1) is the product of two consecutive integers and so it is always an even integer.
Thus, asking whether x^2+x+y = x(x+1) + y is odd is equivalent to asking whether y is odd.

Therefore, B is the answer.
Answer: B