For consecutive integers x and y , where xy , what is the value of

Question

Tricky question.

For consecutive integers  x  and  y , where  x>y , what is the value of  x ?

(1)  x^2−y^2=37 
(2)  x^2+y^2=685

BrentGMATPrepNow · Accepted Answer

Target question :    What is the value of x?

Given : x and y are CONSECUTIVE integers, and x > y   
Since x and y are consecutive integers, and since x is bigger than y, we can conclude that y is ONE LESS than x. 
In other words,  y = x - 1

Statement 1 : x² - y² = 37   
Take this equation, and replace y with  x - 1  to get: x² - ( x - 1 )² = 37
Expand: x² - (x² - 2x + 1) = 37
Simplify: 2x - 1 = 37
Solve to get:  x = 19 
PERFECT!
Since we can answer the  target question  with certainty, statement 1 is SUFFICIENT

Statement 2 : x² + y² = 685  
Take this equation, and replace y with  x - 1  to get: x² + ( x - 1 )² = 685
Expand: x² + (x² - 2x + 1) = 685
Simplify: 2x² - 2x + 1 = 685
Subtract 685 from both sides to get: 2x² - 2x - 684 = 0
Divide both sides by 2 to get: x² - x - 342 = 0
Factor to get: (x - 19)(x + 18) = 0
So, EITHER  x = 19  OR  x = -18 
Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

ASIDE: Notice that  x = 19  and  x = -18  both satisfy statement 2. 
If x = 19, then y = 18, in which case x² + y² = 19² + 18² = 685
If x = -18, then y = -19, in which case x² + y² = (-18)² + (-19)² = 685

Answer: A

Cheers,
Brent

lorenzo393 · Answer

I think that once you get at statement two you get at :2x² - 2x - 684 = 0 you can easily stop you can see immediately that the argument inside the sqrt will be positive--> 2 solutions

BillyZ · Answer

Official solution from Veritas Prep.

This problem rewards two important GMAT concepts. One relates directly to Data Sufficiency as a question type, for which you want to always Leverage Assets; here that means taking the definition "consecutive integers" and turning that into a mathematical operation  (x−y=1) . The other relates to fluency with the Common Algebraic Equations, and in particular here the Difference of Squares equation:

x^2−y^2=(x+y)(x−y)

Once you see that Statement 1 fits the Difference of Squares, you should immediately factor it so that you see the equation both ways.

x^2−y^2=37

Also means that:

(x+y)(x−y)=37

If you then pair that with the equation  x−y=1 , which you derived from the definitions "consecutive integers" and  x>y,  you then know that:  1(x+y)=37 . And for two consecutive integers to sum to 37, that must mean that  x=19  and  y=18 . Therefore, you have sufficient information with Statement 1 to determine that  x=19 . With Statement 1, you can also leverage the information  x−y=1 , manipulation that equation to know that  y=x−1 . That allows you to get close, substituting that into the given equation:

x^2+(x−1)^2=685

That then allows you to expand the parenthetical:

x^2+x^2−2x+1=685

And then combine like terms:

2x^2−2x=684

Then divide by 2:

x^2−x=342

By this point, you should see that this quadratic will lead to multiple solutions for x (those solutions are 19 and −18), meaning that Statement 2 is not sufficient. Note that you can save those algebraic steps if you test the numbers you've already identified as possible consecutive integers from statement 1. 192 and 182 will sum to 685, but since both terms are squares the consecutive integer pairing will work with both positive numbers and negative numbers, so you cannot determine which pair from this statement alone).

GMATBusters · Answer

(1)  x^2−y^2=37 
(x-y)(x+y)= 37
As x>y and x and y are consecutive integer, x-y = 1
(x+y)= 37
x+x-1= 37
2x=38
x= 19
Sufficient

(2)  x^2+y^2=685 
It is very interesting to analyze St2

As it is sum of square of x and y, we can straightaway say that we will not be able to find a unique of x and y
as the values can be negative or positive.

Taking a simple example
  4^2+3^2=25
(-3)^2+(-4)^2=25

Statement 2 is NOT sufficient

Answer A

For consecutive integers x and y , where x>y , what is the value of

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions