If x and y are distinct positive integers, is |x − y| a factor of 12?

Question

(1)  x^2 − 6x + y^2 − 4y = 0

(2) x = 1

chetan2u · Accepted Answer

Great question  Bunuel .. 
Noone has answered in last 3 hours shows it is tricky..

(1)  x^2 − 6x + y^2 − 4y = 0 
  x^2 − 6x + y^2 − 4y = 0........x^2-6x+9+y^2-4y+4=9+4......(x-3)^2+(y-2)^2=13=4+9 
Only possible values is 4+9 as we have to find 13 as sum of two squares - 4+9
 Two ways 
a)  x-3=x-3=√9=3......x=6  AND  y-2=√4=2 ..... y=4  therefore  |x-y|=6-4=2 ..yes a factor of 12
b)  x-3=√4.....x=5  AND  y-2=√9=3...y=5  ...BUT x and y are distinct.. so not possible
Ans yes 
Sufficient

(2) x = 1
Nothing about y 
Insufficient

A

ShankSouljaBoi · Answer

E

1 ---->
x(x-6) + y(y-6) = 0
four cases

x=0 y= 0 No
x=6 y =0 Yes
x=0 y=4 Yes
x= 6 y = 4 Yes
Insuff.

2. x=1 Insuffcient
1+2----. x=1 that means y = -1 or 5
----> mod(x-y) = 0 or 4 
Insufficient.

Please point out my mistake.

ShankSouljaBoi · Answer

Hi !

How did you come to sum of squares will equal 13 ?

jaysonbeatty12 · Answer

This is definitely a question where the math is pretty interesting. If you are a math person, the danger here, strategically, is getting sucked into the fun of the math and wasting three minutes. If you are not a math person, the danger here is staring at statement one for three minutes and end up guessing in a panic.

So, this is how I recommend approaching a question like this. First, evaluate the easier statement first. Statement two is NOT sufficient, so eliminate B and D.

Second, set yourself a time budget to evaluate the harder statement. Whatever strategy you choose - math or example numbers being (probably) the two primary strategies - can take a lot of time and you don't want to spend too much time choosing between A, C, and E.

For statement one this is what I would do. First, I would rearrange the equation to look like this:

x(x-6) + y(y-4) = 0

There are two ways that this statement could equal zero.

1. Both parts equal 0 meaning x(x-6)=0 and y(y-4)=0
2. One is positive and the other is negative but with the same absolute value.

Take the easier of the two situations first. If this is the case, x = 6 and y = 4, so |x - y| = |6-4| = 2, which is a factor of 12. So, statement one works at least sometimes.

By this time, you've likely spent at least 2 minutes. Working through 2 (one positive and one negative) is going to take a lot of math or example numbers. DON'T DO THIS. At this point, guess among A, C, and E and move on.

If you are really ahead on time, the arithmetic is probably easier than the math. Make a table and pick some numbers to see what works. I came up with

x y
1 5
6 4 (from above)

This list may not be exhaustive, but for the purposes of the GMAT it is probably fine. For these, |x-y| is a factor of 12. Go with A and move on.

chetan2u · Answer

Firstly the MISTAKE ...
x and y are   distinct    +   positive   +  integer  
now your values..
in three cases x or y or both are 0 but 0 is not positive so discard those pairs..
ONLY 6,4 are left....
hence A is suff

NOW on your question how square are 13
 x^2 − 6x + y^2 − 4y = 0........x^2-6x+9+y^2-4y+4=9+4......(x-3)^2+(y-2)^2=13=4+9 
x^2-6x is of the form a^2-2ab+b^2=(a-b)^2 so a is x, b becomes 3.... x^2-2*3*x+3^2=(x-3)^2  so we add 3^2 on BOTH sides and similarly 2^2 with y^2-4y
combined 9+4 on both sides
Only possible values is 4+9 as we have to find 13 as sum of two squares - 4+9

ShankSouljaBoi · Answer

Thank you !! What a blunder

StudiosTom · Answer

Experts,I want some suggestion from you on this approach:
Since statement 2 seem to be simple i used that and substituted it in statement 1.Statement 1 involved 2 variables and 1 equation and so would have many possible roots.I did not add 9 and 4 as chetan2u has done as that did not occur to me.By solving for x=1 ,i get y=5 or y=1.Since distinct and positive y=5.
and |x-y|=4,so C answer.

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If x and y are distinct positive integers, is |x − y| a factor of 12?

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If x and y are distinct positive integers, is |x − y| a factor of 12?

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