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# ‎[x] denotes to be the least integer no less than x. Is [2d]

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Manager
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‎[x] denotes to be the least integer no less than x. Is [2d] [#permalink]  17 Apr 2012, 18:53
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Question Stats:

56% (02:31) correct 43% (00:54) wrong based on 109 sessions
‎[x] denotes to be the least integer no less than x. Is [2d] = 0?

(1) [d] = 0
(2) [3d] = 0
[Reveal] Spoiler: OA
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Kudos [?]: 33 [1] , given: 34

Re: Integers [#permalink]  17 Apr 2012, 19:49
1
KUDOS
Smita04 wrote:
‎[x] denotes to be the least integer no less than x. Is [2d] = 0?
(1) [d] = 0
(2) [3d] = 0

'no less than' indicated [x]>= x....with the equality holding when x is an integer

st1: [d]=0 indicates -1<d<0 (eg. -0.5) => -2<2d<0(2x -0.5=-1) =>[2d]=-1
but if d= -0.3......2d= -0.6.....[2d]= 0 (not -1 as -1< -0.6)
NOT Sufficient

st2: -1<3d<0 => -1/3<d<0
d can be -0.65...then [2d]=0..... d can be [-0.01]....[d]=0....
(taking extreme values )
Suffecient

Last edited by Dreaming on 17 Apr 2012, 22:43, edited 1 time in total.
Math Expert
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Kudos [?]: 17576 [1] , given: 2225

Re: ‎[x] denotes to be the least integer no less than x. Is [2d] [#permalink]  17 Apr 2012, 20:17
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KUDOS
Expert's post
‎[x] denotes to be the least integer no less than x. Is [2d] = 0?

[x] denotes to be the least integer no less than x, means that some function [] rounds UP a number to the nearest integer, for example:
[1.5] = 2 since 2 is the least integer which no less than 1.5;
[-0.5]=0 since 0 is the least integer which no less than -0.5;
[1] = 1 since 1 itself is the least integer which no less than 1;
...

(1) [d] = 0 --> -1<d\leq{0}. Now, if d=0 then [2d]=[0]=0 but if d=-0.5 then [2d]=[-1]=-1. Not sufficient.

Or: -1<d\leq{0} --> -2<d\leq{0} --> [2d]=0 (if -1<2d\leq{0}) or [2d]=-1 (if -2<2d\leq{-1}). Not sufficient.

(2) [3d] = 0 --> -1<3d\leq{0} --> -\frac{1}{3}<d\leq{0}. Even if d=-\frac{1}{3} then [2d]=[-\frac{2}{3}]=0 (again since 0 is the least integer which no less than -2/3). Sufficient.

Or: -1<3d\leq{0} --> -\frac{2}{3}<2d\leq{0} --> [2d]=0. Sufficient.

Similar questions to practice:
if-denotes-the-greatest-integer-less-than-or-equal-to-x-94687.html
for-all-z-denotes-the-least-integer-greater-than-or-110561.html

Hope it helps.
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Math Expert
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Kudos [?]: 17576 [0], given: 2225

Re: ‎[x] denotes to be the least integer no less than x. Is [2d] [#permalink]  21 Jun 2013, 01:51
Expert's post
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

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Kudos [?]: 8 [0], given: 7

Re: ‎[x] denotes to be the least integer no less than x. Is [2d] [#permalink]  22 Jun 2013, 02:56
Smita04 wrote:
‎[x] denotes to be the least integer no less than x. Is [2d] = 0?

(1) [d] = 0
(2) [3d] = 0

I did not jump into the equations, but tried to solve logically. Please let me know if wrong.

We need to find out if [2d] = 0?

Again its a YES or NO, question, getting Either in the statements will help us to the solutions.

(1) [d] = 0

Now according to the question [x] = integer no less than x

Hence, considering two extremes for evaluating this particular case, Lets take
[0.12] = 0 as well as [0.99] = 0 (both are valid for [d] =0)

therefore, [2d], i.e. taking the same values as above
[0.24] = 0, but [1.98] = 1
so we cant say most definitely that [2d] =0

Not Sufficient.

(2) [3d] = 0

again considering etremes here,

lets take
Minimum value => 3d = 0.12 (you can take less then this as well, i am just taking this for easy divisibility purpose, as anything less then 0.12 will also result in 0 we already know that)
and
Maximum value => 3d = 0.99
[3d] = 0 holds for both the values.

now, [2d]
minimum= [0.8] =0
maximum= [0.22] =0

hence, no matter what range we take for 3d, 2d will always be less then that.
Thus, if [3d]= 0, holds true, then [2d] =0 should hold true as well.

Hence, B
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Re: ‎[x] denotes to be the least integer no less than x. Is [2d] [#permalink]  23 Jun 2013, 05:53
Bunuel wrote:
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

All DS Functions and Custom Characters questions: search.php?search_id=tag&tag_id=40
All PS Functions and Custom Characters questions: search.php?search_id=tag&tag_id=61

Hi Bunuel,

Please let me know if the above approach is correct?
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Posts: 16817
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Kudos [?]: 17576 [0], given: 2225

Re: ‎[x] denotes to be the least integer no less than x. Is [2d] [#permalink]  24 Jun 2013, 00:59
Expert's post
kpali wrote:
Smita04 wrote:
‎[x] denotes to be the least integer no less than x. Is [2d] = 0?

(1) [d] = 0
(2) [3d] = 0

I did not jump into the equations, but tried to solve logically. Please let me know if wrong.

We need to find out if [2d] = 0?

Again its a YES or NO, question, getting Either in the statements will help us to the solutions.

(1) [d] = 0

Now according to the question [x] = integer no less than x

Hence, considering two extremes for evaluating this particular case, Lets take
[0.12] = 0 as well as [0.99] = 0 (both are valid for [d] =0)

therefore, [2d], i.e. taking the same values as above
[0.24] = 0, but [1.98] = 1
so we cant say most definitely that [2d] =0

Not Sufficient.

(2) [3d] = 0

again considering etremes here,

lets take
Minimum value => 3d = 0.12 (you can take less then this as well, i am just taking this for easy divisibility purpose, as anything less then 0.12 will also result in 0 we already know that)
and
Maximum value => 3d = 0.99
[3d] = 0 holds for both the values.

now, [2d]
minimum= [0.8] =0
maximum= [0.22] =0

hence, no matter what range we take for 3d, 2d will always be less then that.
Thus, if [3d]= 0, holds true, then [2d] =0 should hold true as well.

Hence, B

No, that's not correct. Function [] rounds UP a number to the nearest integer, thus [0.12] = 1, not 0 and [0.99] = 1, not 1.
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Re: ‎[x] denotes to be the least integer no less than x. Is [2d]   [#permalink] 24 Jun 2013, 00:59
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