How many odd integers are greater than the integer x and

You are genius. Many thanks Bunuel

hey bunnel, when you say "4 integers more than X=1 and less than Y=6" why cant it be 1,2,2,2,2,6, or, 1,2,5,5,5,6 this satisfies the statement rather than consecutive integers, right, the question did not say only consecutive or different numbers,,,, ,,iam i missing somethi here,,,,could you explain

What is "1,2,2,2,2,6, or, 1,2,5,5,5,6"?

Let me ask you a question: how many integers are more than 1 and less than 6? Can the answer be other than 4 (2, 3, 4 and 5)?

Or another way: if 1<n<6 then how many integer values n can take?

The first statement says there are 12 even integers. There are two cases for this:

1. x is an odd integer, in which case there will be 12 even integers but only 11 odd integers.
2. x is an even integer, in which case there will be 12 each of even and odd integers.

Hence this is insufficient.

Statement 2 says there are 24 integers greater than x and lesser than y.

Two cases:

1. x is an odd integers, which means there will be 12 even integers and 12 odd integers.
2. x is an even integer, in which case there will be 12 each of even and odd integers.

In both cases, the answer is the same and hence answer is B. I got tripped up by this question, good one.

Bunuel, I chose E instead of B, since it is not stated that the integers are consecutive, so do you have any explanation on this matter ?
thank you

(2) says that: there are 24 integers greater than x and less than y. Naturally those 24 integers between x and y are consecutive, how else? Consider x=1 and y=26: there are following 24 integers between them: 2, 3, 4, ..., 25.

Also check this post about the same issue: how-many-odd-integers-are-greater-than-the-integer-x-and-100521.html#p809821

Hope it's clear.

I think the best way to cehck 2 would be to reduce the number to 4 from 24 and try out.. eoeoeo.. the result would be in sync with 24 numbers .. same applies to 1 as well.

what say?

Yes that should work for this problem. Even I would have solved this way (reduce the sample size from 12 to 2 and 24 to 4)
1) There are 2 even integers greater than x and less than y
INSUFFICIENT: XeoeY or XoeoeY - gives 1 or 2 odds
2) There are 4 integers greater than x and less than y
SUFFICIENT: XoeoeY or XeoeoY - both give same number of odds, 2. If it works for 4 then it will work for 24.

Hence choice(B).

hi bunnel

st 1 can still be written and tried...
but st 2 will be time consuming... how do we know for sure that when there r 24 even I between x & y then there will be 24 odd I, without manually doing this? as doing it manually time consuming...

as you suggest we should take smaller number to try but how to decide which numbers will gave same result as 12 & 24..
any double? like i can take 4 for case 1 and 8 for case 2?

(2) says that there are 24 (even) integers greater than integer x and less than integer y. The important part is that the number of integers between x and y is even. In this case half of them must be odd and another half must be even. How else? Can there be 11 odd integers and 13 odd integer greater than x and less than y?

If we were told that there are 3 (odd) integers greater than integer x and less than integer y, then this would be insufficient, because there could be 1 odd and 2 evens or 2 odds and 1 even.

Target question: How many odd integers are greater than the integer x and less than the integer y?

Statement 1: There are 12 even integers greater than x and less than y
There are many scenarios that satisfy statement 1. Here are two:
Case a: x = 1 and y = 25. In this case, the answer to the target question is there are 11 odd integers greater than the integer x and less than the integer y
Case b: x = 1 and y = 26. In this case, the answer to the target question is there are 12 odd integers greater than the integer x and less than the integer y
Since we can’t answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: There are 24 integers greater than x and less than y
ASIDE: Many students will read the statement 2 and conclude that, since we aren't told whether those 24 integers are CONSECUTIVE integers, there's no way to tell how many how many odd integers there are.
However, the integers between x and y will always be consecutive.
For example, the integers between 5 and 11 are 6, 7, 8, 9 and 10 (consecutive)
Similarly, the integers between -2 and 8 are -1, 0, 1, 2, 3, 4, 5, 6 and 7 (consecutive)

Since even and odd integers alternate in consecutive integers (e.g., 1 is odd, 2 is even, 3 is odd, 4 is even,... etc.) , we know that 12 of the 24 integers must be even, and the other 12 must be odd
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

Answer: Option B

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IanStewart please explain this question? In a simpler way

I haven't read the other explanations, but looking first at Statement 2:

(2) There are 24 integers greater than x and less than y

Since even and odd integers alternate, 12 of these integers must be even, and the other 12 must be odd. So Statement 2 is sufficient. If instead Statement 2 said "There are 25 integers between x and y" (or any other odd number), it would not be sufficient, because it might be that 13 are even, 12 are odd, or it might be that 13 are odd and 12 are even.

Looking at the harder Statement, Statement 1:

(1) There are 12 even integers greater than x and less than y

It doesn't matter which integers these are, so we can just imagine they are the twelve integers 2, 4, 6, 8, ..., 22, 24. These need to be the only even integers between the two integers x and y. So it might be that x = 1 and y = 25, and then there are eleven odd integers between x and y. But it might be that x = 1 and y = 26, in which case we'll have one more odd integer in our range, namely 25, so we could have twelve odd integers in total. Or it might even be that x = 0 and y = 26, in which case we'll also have '1' in our range, and we'll have thirteen odd integers between x and y. So from Statement 1, all we know is that there are 11, 12 or 13 odd integers between x and y, and Statement 1 is not sufficient. So the answer is B.

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