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Math Expert V
Joined: 02 Sep 2009
Posts: 60460
Re: How many odd integers are greater than the integer x and  [#permalink]

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Bunuel wrote:
Orange08 wrote:
How many odd integers are greater than the integer x and less than the integer y?

1) There are 12 even integers greater than x and less than y
2) There are 24 integers greater than x and less than y

Sorry, I tried to search the forum for previous explanations. But since the search was too generic, it didn't fetch any results.

(1) Here is the string of 12 even integers and 11 odd integers between them: eoeoeoeoeoeoeoeoeoeoeoe.

4 cases are possible:

If x and y are both odd then XeoeoeoeoeoeoeoeoeoeoeoeY then there will be 11 odd integers between them;
If x and y are both even then XoeoeoeoeoeoeoeoeoeoeoeoeoY then there will be 11+2=13 odd integers between them;
If x is even and y is odd then XoeoeoeoeoeoeoeoeoeoeoeoeY then there will be 11+1=12 odd integers between them;
If x is odd and y is even then XeoeoeoeoeoeoeoeoeoeoeoeoY then again there will be 11+1=12 odd integers between them.

Not sufficient.

(2) Out of 24 consecutive integers greater than X and less than Y in any case half will be odd and another half will be even, so there are 12 odd integers greater than X and less than Y. Sufficient.

P.S. You can try instead of 12 and 24 some smaller numbers to simplify calculations for example 2 even integers for (1) and 4 integers for (2).

Orange08 wrote:
why are the integers assumed consecutive over here?

Are you talking about (2)? If we are told that there are 4 integers more than X=1 and less than Y=6, then these integers are 2, 3, 4, and 5 - 4 consecutive integers, how else? Note that half are odd and half are even. Or if X=2 and Y=7 then these integers would be 3, 4, 5 and 6 - 4 consecutive integers: half are odd and half are even.

Hope it's clear.

Hi Bunuel
If in statement-2, there is odd number ..like 5 integers between x and y then it will be insuff ?
Thanks.

Yes, in that case the statement would be insufficient. We could have 2 odd, 3 even or 2 even, 3 odd.
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Re: How many odd integers are greater than the integer x and  [#permalink]

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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

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

(1) There are 12 even integers greater than x and less than y
(2) There are 24 integers greater than x and less than y

There are 2 variables (x,y) and 2 equations from the question and the 2 conditions, so there is high chance (C) will be our answer.
Looking at the conditions together, if there are 12 even numbers out of the 24 integers, there are of course 12 odd integers, so the answer becomes (C). But this is an integer question which is one the the key questions, if we apply 4(A) mistake types,
Looking at condition 1, the number of odd integers becomes (35-11)/2+1=13 when x=10, y=36, and (33-13)/2+1=11 when x=11, y=35; this does not give unique answer, so this is insufficient.
From condition 2, if there are 24 integers, there has to be 12 even and 12 odd. This is sufficient, making the answer (B).

For cases where we need 2 more equation, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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Re: How many odd integers are greater than the integer x and  [#permalink]

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Bunuel wrote:
Orange08 wrote:
How many odd integers are greater than the integer x and less than the integer y?

1) There are 12 even integers greater than x and less than y
2) There are 24 integers greater than x and less than y

Sorry, I tried to search the forum for previous explanations. But since the search was too generic, it didn't fetch any results.

(1) Here is the string of 12 even integers and 11 odd integers between them: eoeoeoeoeoeoeoeoeoeoeoe.

4 cases are possible:

If x and y are both odd then XeoeoeoeoeoeoeoeoeoeoeoeY then there will be 11 odd integers between them;
If x and y are both even then XoeoeoeoeoeoeoeoeoeoeoeoeoY then there will be 11+2=13 odd integers between them;
If x is even and y is odd then XoeoeoeoeoeoeoeoeoeoeoeoeY then there will be 11+1=12 odd integers between them;
If x is odd and y is even then XeoeoeoeoeoeoeoeoeoeoeoeoY then again there will be 11+1=12 odd integers between them.

Not sufficient.

(2) Out of 24 consecutive integers greater than X and less than Y in any case half will be odd and another half will be even, so there are 12 odd integers greater than X and less than Y. Sufficient.

P.S. You can try instead of 12 and 24 some smaller numbers to simplify calculations for example 2 even integers for (1) and 4 integers for (2).

Orange08 wrote:
why are the integers assumed consecutive over here?

Are you talking about (2)? If we are told that there are 4 integers more than X=1 and less than Y=6, then these integers are 2, 3, 4, and 5 - 4 consecutive integers, how else? Note that half are odd and half are even. Or if X=2 and Y=7 then these integers would be 3, 4, 5 and 6 - 4 consecutive integers: half are odd and half are even.

Hope it's clear.

I might sound dumb but the fact that consecutive is confusing. we're only told 24 integers are greater than x :/
I was confused between B and E
Math Expert V
Joined: 02 Sep 2009
Posts: 60460
Re: How many odd integers are greater than the integer x and  [#permalink]

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Bunuel wrote:
Orange08 wrote:
How many odd integers are greater than the integer x and less than the integer y?

1) There are 12 even integers greater than x and less than y
2) There are 24 integers greater than x and less than y

Sorry, I tried to search the forum for previous explanations. But since the search was too generic, it didn't fetch any results.

(1) Here is the string of 12 even integers and 11 odd integers between them: eoeoeoeoeoeoeoeoeoeoeoe.

4 cases are possible:

If x and y are both odd then XeoeoeoeoeoeoeoeoeoeoeoeY then there will be 11 odd integers between them;
If x and y are both even then XoeoeoeoeoeoeoeoeoeoeoeoeoY then there will be 11+2=13 odd integers between them;
If x is even and y is odd then XoeoeoeoeoeoeoeoeoeoeoeoeY then there will be 11+1=12 odd integers between them;
If x is odd and y is even then XeoeoeoeoeoeoeoeoeoeoeoeoY then again there will be 11+1=12 odd integers between them.

Not sufficient.

(2) Out of 24 consecutive integers greater than X and less than Y in any case half will be odd and another half will be even, so there are 12 odd integers greater than X and less than Y. Sufficient.

P.S. You can try instead of 12 and 24 some smaller numbers to simplify calculations for example 2 even integers for (1) and 4 integers for (2).

Orange08 wrote:
why are the integers assumed consecutive over here?

Are you talking about (2)? If we are told that there are 4 integers more than X=1 and less than Y=6, then these integers are 2, 3, 4, and 5 - 4 consecutive integers, how else? Note that half are odd and half are even. Or if X=2 and Y=7 then these integers would be 3, 4, 5 and 6 - 4 consecutive integers: half are odd and half are even.

Hope it's clear.

I might sound dumb but the fact that consecutive is confusing. we're only told 24 integers are greater than x :/
I was confused between B and E

(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.
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Re: How many odd integers are greater than the integer x and  [#permalink]

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Bunuel wrote:
Orange08 wrote:
How many odd integers are greater than the integer x and less than the integer y?

1) There are 12 even integers greater than x and less than y
2) There are 24 integers greater than x and less than y

Sorry, I tried to search the forum for previous explanations. But since the search was too generic, it didn't fetch any results.

(1) Here is the string of 12 even integers and 11 odd integers between them: eoeoeoeoeoeoeoeoeoeoeoe.

4 cases are possible:

If x and y are both odd then XeoeoeoeoeoeoeoeoeoeoeoeY then there will be 11 odd integers between them;
If x and y are both even then XoeoeoeoeoeoeoeoeoeoeoeoeoY then there will be 11+2=13 odd integers between them;
If x is even and y is odd then XoeoeoeoeoeoeoeoeoeoeoeoeY then there will be 11+1=12 odd integers between them;
If x is odd and y is even then XeoeoeoeoeoeoeoeoeoeoeoeoY then again there will be 11+1=12 odd integers between them.

Not sufficient.

(2) Out of 24 consecutive integers greater than X and less than Y in any case half will be odd and another half will be even, so there are 12 odd integers greater than X and less than Y. Sufficient.

P.S. You can try instead of 12 and 24 some smaller numbers to simplify calculations for example 2 even integers for (1) and 4 integers for (2).

Orange08 wrote:
why are the integers assumed consecutive over here?

Are you talking about (2)? If we are told that there are 4 integers more than X=1 and less than Y=6, then these integers are 2, 3, 4, and 5 - 4 consecutive integers, how else? Note that half are odd and half are even. Or if X=2 and Y=7 then these integers would be 3, 4, 5 and 6 - 4 consecutive integers: half are odd and half are even.

Hope it's clear.

Great explanation as always Buunel! Happy to see the consecutive integers doubt clarified as well.

A little off topic, but first look at eoeoeoeoeoeoeoeoeoeoeoe reminded me of the minion from Despicable Me Manager  B
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Posts: 64
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Re: How many odd integers are greater than the integer x and  [#permalink]

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Bunuel wrote:
Orange08 wrote:
How many odd integers are greater than the integer x and less than the integer y?

1) There are 12 even integers greater than x and less than y
2) There are 24 integers greater than x and less than y

Sorry, I tried to search the forum for previous explanations. But since the search was too generic, it didn't fetch any results.

(1) Here is the string of 12 even integers and 11 odd integers between them: eoeoeoeoeoeoeoeoeoeoeoe.

4 cases are possible:

If x and y are both odd then XeoeoeoeoeoeoeoeoeoeoeoeY then there will be 11 odd integers between them;
If x and y are both even then XoeoeoeoeoeoeoeoeoeoeoeoeoY then there will be 11+2=13 odd integers between them;
If x is even and y is odd then XoeoeoeoeoeoeoeoeoeoeoeoeY then there will be 11+1=12 odd integers between them;
If x is odd and y is even then XeoeoeoeoeoeoeoeoeoeoeoeoY then again there will be 11+1=12 odd integers between them.

Not sufficient.

(2) Out of 24 consecutive integers greater than X and less than Y in any case half will be odd and another half will be even, so there are 12 odd integers greater than X and less than Y. Sufficient.

P.S. You can try instead of 12 and 24 some smaller numbers to simplify calculations for example 2 even integers for (1) and 4 integers for (2).

Orange08 wrote:
why are the integers assumed consecutive over here?

Are you talking about (2)? If we are told that there are 4 integers more than X=1 and less than Y=6, then these integers are 2, 3, 4, and 5 - 4 consecutive integers, how else? Note that half are odd and half are even. Or if X=2 and Y=7 then these integers would be 3, 4, 5 and 6 - 4 consecutive integers: half are odd and half are even.

Hope it's clear.

In statement 2- If the number of integers between X and Y was odd, say 23 or 21. Would the statement be sufficient?
What would have been the answer in that case?
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Re: How many odd integers are greater than the integer x and  [#permalink]

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

(1) There are 12 even integers greater than x and less than y
(2) There are 24 integers greater than x and less than y

1.
let just say there are 3 even integers greater than x and less than y

if x =even
x odd e odd e odd e odd => so 4 odd.
if X =odd
o e o e o e o => 2 odd

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

let just say there are 4 integers greater than x and less than y

e o e o e 0 => 2 odd
o e o e 0 o => 2 odd

THis is working only if There are 24 integers greater than x and less than y, the value in red is even. if it is odd. will not work. (untill unless we know what x is even /odd)
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Re: How many odd integers are greater than the integer x and  [#permalink]

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sudhirgupta93 wrote:
Bunuel wrote:
Orange08 wrote:
How many odd integers are greater than the integer x and less than the integer y?

1) There are 12 even integers greater than x and less than y
2) There are 24 integers greater than x and less than y

Sorry, I tried to search the forum for previous explanations. But since the search was too generic, it didn't fetch any results.

(1) Here is the string of 12 even integers and 11 odd integers between them: eoeoeoeoeoeoeoeoeoeoeoe.

4 cases are possible:

If x and y are both odd then XeoeoeoeoeoeoeoeoeoeoeoeY then there will be 11 odd integers between them;
If x and y are both even then XoeoeoeoeoeoeoeoeoeoeoeoeoY then there will be 11+2=13 odd integers between them;
If x is even and y is odd then XoeoeoeoeoeoeoeoeoeoeoeoeY then there will be 11+1=12 odd integers between them;
If x is odd and y is even then XeoeoeoeoeoeoeoeoeoeoeoeoY then again there will be 11+1=12 odd integers between them.

Not sufficient.

(2) Out of 24 consecutive integers greater than X and less than Y in any case half will be odd and another half will be even, so there are 12 odd integers greater than X and less than Y. Sufficient.

P.S. You can try instead of 12 and 24 some smaller numbers to simplify calculations for example 2 even integers for (1) and 4 integers for (2).

Orange08 wrote:
why are the integers assumed consecutive over here?

Are you talking about (2)? If we are told that there are 4 integers more than X=1 and less than Y=6, then these integers are 2, 3, 4, and 5 - 4 consecutive integers, how else? Note that half are odd and half are even. Or if X=2 and Y=7 then these integers would be 3, 4, 5 and 6 - 4 consecutive integers: half are odd and half are even.

Hope it's clear.

In statement 2- If the number of integers between X and Y was odd, say 23 or 21. Would the statement be sufficient?
What would have been the answer in that case?

Answer would be insufficient - eg. take smaller nos.
if x=1 & y=5, so we have 1 <2,3,4<5 so only 1 odd no
if x=2 & y=6, so we have 2<3,4,5<6 , so now we have 2 odd nos.

I have posted a similar query to bunuel for clarification regarding generalization of this counting principle.But from what I can gather, when we have an even no of consecutive integers, we will have have an equal no of odd & equal no of even nos between them. & when we have even no or odd number of integers to count, then it will depend on the 2 numbers encapsulating them.
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Re: How many odd integers are greater than the integer x and  [#permalink]

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

What if;

1<N<1000----> Any 24 numbers between 1 and 1000 will also satisfy the second argument of the question.

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Re: How many odd integers are greater than the integer x and  [#permalink]

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20043856 wrote:
2) There are 24 integers greater than x and less than y

What if;

1<N<1000----> Any 24 numbers between 1 and 1000 will also satisfy the second argument of the question.

Hello

If you are taking X=1 and Y=1000, then there are not just '24' integers between them. Between 1 and 1000 there are '998' integers (2, 3, 4, ...999). So this example (of 1 and 1000) cannot be taken considering the second statement.

When the statement says, "there are 24 integers between X and Y", it means there are '24 integers only'. That is only possible if Y = X+25. And out of those 24 integers, 12 will be odd and 12 will be even (no matter whether X is odd or even, or whether Y is even or odd)
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How many odd integers are greater than the integer x and  [#permalink]

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3
nothingman wrote:
Hi! why is it that nobody (and I mean on any forum) has considered X to be a negative integer? If that's the case then the option E would be the clear answer, wouldn't it?

It doesnt matter whether x or y are negative, the pattern will still be the same. Note that 0 is considered as even, even though it is neither negative nor positive.
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GMAT 1: 670 Q46 V36 GMAT 2: 690 Q47 V38 Re: How many odd integers are greater than the integer x and  [#permalink]

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Orange08 wrote:
How many odd integers are greater than the integer x and less than the integer y?

(1) There are 12 even integers greater than x and less than y
(2) There are 24 integers greater than x and less than y

Method: Solving an easier Q and testing cases

1)
Let x = 1, y = 5
#s in middle: 2,3,4 => # of Odd ints = 1 (i.e. 3)
Let x = 1, y = 6
#s in middle: 2,3,4,5 => # of Odd ints = 2 (i.e. 2)
Not sufficient

2)
Let x = 1, y = 6
#s in middle: 2,3,4,5 => # of Odd ints = 2 (i.e. 3,5)
Let x = 1, y = 7
#s in middle: 2,3,4,5,6 => # of Odd ints = 2 (i.e. 3,5)
Sufficient

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Re: How many odd integers are greater than the integer x and  [#permalink]

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I do not still get it...because it does not state that the numbers are consecutive....

(2) There are 24 integers greater than x and less than y. <- who knows these 24 integers are all odd numbers or even numbers without saying these are consecutive numbers...
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How many odd integers are greater than the integer x and  [#permalink]

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Hi,Thanks for the explanations above but in Statement 2 why didn't we consider the order of the integers as such
{x,0(neither odd nor even ),1(odd),2(Even),....23,y }there are 24 integers here also. the question did not mention positive integers right? How many odd integers are greater than the integer x and   [#permalink] 13 Dec 2019, 02:46

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