GMAT Prep - Interger Question

[dongkim2]

How many odd intergers are greater than the interger X and less than the interger Y?

(1) There are 12 even intergers greater than X and less than Y.

(2) There are 24 intergers greater than X and less than Y.

OA: B


(1)
If X=1 and Y=25 (X=1< 2,4,~~~,24 <Y=25; 12 even intergers), then you can get 11 odd intergers.
If X=0 and Y=26 (X=0< 2,4,~~~,24 <Y=26; 12 even intergers), then you can get 13 odd intergers.
Therefore, A is not sufficient.

(2)
If X=1<2,3,4,~~~,25<Y=26, then you can get 12 odd intergers.
If X=2<3,4,5,~~~,26<Y=27, Then you can get 12 odd intergers.
Therefore, B is sufficient.

Can anyone please comment on my approach? Is this okay? If there is any better way to approach this question, please let me know. Thanks!

[mj78ind]

That is how i did it, with a slight twist rather than going for 24 or 12 number long integer lists you could solve with 3 - 4 number long integer lists which would mirror the 25 integers.

[Patrick_GMATFix]

hi Dongkim. I think your way is fine. Below is the logic behind the right answer.

STATEMENT 2: Essentially, because integers always alternate between even and odd, any set of consecutive integers that contains an even number of values must have the same number of evens and odds. Statement (2) is sufficient because a set of 24 consecutive integers must have 12 evens and 12 odds.

STATEMENT 1: If a set of consecutive integers has 12 evens, it may have 12 odds, but it may also have 11 odds (if we start and end with an even, we will have one more even than odd). In fact, this set could have 13 odds (if we start and end with an odd, we will have one more odd than even). This is why statement 1 does not tell us exactly how many odds there are. Thus statement 1 is not sufficient.

The answer is B

A more detailed solution as well as a video solution can be found at GMATPrep Question 1228. To practice similar questions, set topic='Number Properties' and difficulty='600-700' in the Drill Generator.

Have a good one,
-Patrick

[MBA]

Hi Patrick,

I have two points,please suggest the explanation

1)here the problem says X & Y are integers,how can we assume it is consecuitive integers?And if the integers between X & Y are not consecutive then even if we know the number of integers between X & Y,it really does not help us.

2)if X & Y are just "integers",then it can be X=-10 & Y=-9.And in that case it does tell us about odd or even integer in between X & Y.

So why the answer is not "E"

[lunarpower]

1)here the problem says X & Y are integers,how can we assume it is consecuitive integers?And if the integers between X & Y are not consecutive then even if we know the number of integers between X & Y,it really does not help us.

this is not a sequence; you're just counting the total number of integers that are between the two values in question. it doesn't actually make any difference whether you count those numbers in order.
for instance, if you want to know how many integers are between 0 and 10, then there are nine of them. it doesn't matter whether you count them in order (1, 2, 3, 4, 5, 6, 7, 8, 9) or out of order (e.g., 8, 5, 9, 2, 1, 7, 4, 6, 3); there are still going to be nine of them.
of course, counting them in any order other than consecutively is going to make the problem about a thousand times as hard as it should be.

2)if X & Y are just "integers",then it can be X=-10 & Y=-9.And in that case it does tell us about odd or even integer in between X & Y.

yeah, but x = -10 and y = -9 don't obey either of the two statements, so they are irrelevant.
which statement(s) were you looking at when you considered these values?

[MBA]

Thanks Ron,

I understood the answer of my first question.

2)if X & Y are just "integers",then it can be X=-10 & Y=-9.And in that case it does tell us about odd or even integer in between X & Y.

For the above mentioned second question,since the original problem states "x and y are integers" so how can we assume that X and Y are positive integers and then go on trying to find out the number of odd and even integers between X & Y.As stated in my 2nd question it can be X=-1 and Y = 22(I changed the numbers to give a relevant exapmple.


Please check my example below,there are 24 integers between X and Y but the number of odd & even integers are different both the scenarios.

Value of X &Y---------Range--------no of odd integer-----------no of even integer
X=-2 and Y=23------ (-1,0,...22 )-------11----------------------------11
X=0 and Y=25------(1,2,....24 )-------12----------------------------12


Thanks

[lunarpower]

Please check my example below,there are 24 integers between X and Y but the number of odd & even integers are different both the scenarios.

Value of X &Y---------Range--------no of odd integer-----------no of even integer
X=-2 and Y=23------ (-1,0,...22 )-------11----------------------------11
X=0 and Y=25------(1,2,....24 )-------12----------------------------12


Thanks

nope. it appears that you are not counting 0, -2, -4, etc., as "even integers", and also not counting -1, -3, -5, etc. as "odd integers".
in fact, 0, -2, -4, etc., are even integers, and -1, -3, -5, etc., are odd integers.

you can prove this to yourself just by realizing that, if these numbers were not considered among the even and odd integers respectively, then basically every rule you've learned about adding and subtracting evens and odds would simply fall apart.
for instance, you know a rule that states "even - even = even". so, 6 - 8 = -2 is even.
and so on.
if you were to needlessly restrict the definition of even and odd to positive integers, then you would need to add caveats to several of these existing rules -- for instance, the "even - even = even" rule would no longer work unless you specify that the first one of these even integers is larger than the second one.

--

still, however, there is one thing in your favor here, which is that i don't think the actual test will require you to know this -- i.e., if this were an official problem, i'm pretty sure that the integers in question would be stated as positive.
this question looks vaguely familiar, so i suspect it might be one of ours (MGMAT). i'll go look in the question database, and, if this is indeed our question, i'll go ahead and add a condition that the integers must be positive.
note that this condition does NOT, at all, change the answer to the problem -- it just makes the problem slightly more like what you'll actually see on exam day.

[MBA]

Thanks a lot

You have really cleared my confusion.

Just FYI,this is GMAT PREP problem.

[lunarpower]

Thanks a lot :-)

You have really cleared my confusion.

Just FYI,this is GMAT PREP problem.

very, very interesting indeed -- so it looks like GMAC really does expect you to know that
* 0, -2, -4, ... are EVEN integers
and that
* -1, -3, -5, ... are ODD integers.

so... know this!