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If [z] denotes the greatest integer less than or equal to z and [z] =  [#permalink]

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Difficulty:   35% (medium)

Question Stats: 67% (01:14) correct 33% (01:14) wrong based on 774 sessions

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If [z] denotes the greatest integer less than or equal to z and [z] = -1 which of the following statements must be true?

A. z = -1
B. -2 <= z < -1
C. -2 < z <= -1
D. -1 <= z < 0
E. -1 < z <= 0

Attachment: prep.JPG [ 68.6 KiB | Viewed 27150 times ]

Originally posted by sguptashared on 23 Oct 2013, 19:50.
Last edited by Bunuel on 06 Nov 2017, 20:51, edited 3 times in total.
Renamed the topic, edited the question, added the OA and moved to PS forum.
Math Expert V
Joined: 02 Sep 2009
Posts: 60647
Re: If [z] denotes the greatest integer less than or equal to z and [z] =  [#permalink]

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10
21
sguptashared wrote:
If [z] denotes the greatest integer less than or equal to z and [z] = -1 which of the following statements must be true?

A. z = -1
B. -2 <= z < -1
C. -2 < z <= -1
D. -1 <= z < 0
E. -1 < z <= 0

[z] is the greatest integer less than or equal to z means that some function [] rounds DOWN a number to the nearest integer. For example [1.5]=1, =2, [-1.5]=-2, ...

Now, since [z] = -1, then $$-1\leq{z}<0$$ --> ANY number from this range when rounded down to the nearest integer is -1.

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Hope it helps.

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##### General Discussion
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Joined: 17 Mar 2014
Posts: 425
Re: If [z] denotes the greatest integer less than or equal to z and [z] =  [#permalink]

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2
Bunuel wrote:
sguptashared wrote:
If [z] denotes the greatest integer less than or equal to z and [z] = -1 which of the following statements must be true?

A. z = -1
B. -2 <= z < -1
C. -2 < z <= -1
D. -1 <= z < 0
E. -1 < z <= 0

[z] is the greatest integer less than or equal to z means that some function [] rounds DOWN a number to the nearest integer. For example [1.5]=1, =2, [-1.5]=-2, ...

Now, since [z] = -1, then $$-1\leq{z}<0$$ --> ANY number from this range when rounded down to the nearest integer is -1.

Similar questions to practice:
if-denotes-the-least-integer-greater-than-or-equal-to-x-132223.html
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for-all-numbers-t-let-t-be-defined-as-the-greatest-105416.html
is-x-an-odd-integer-1-x-2-2x-99856.html
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is-the-greatest-integer-less-than-or-equal-to-the-real-n-154006.html

Hope it helps.

If question asks " [z] denotes the greatest integer less than or equal to z and [z] = 0", then can we say 0<=Z<1 ?

Regards,
Ammu
Math Expert V
Joined: 02 Sep 2009
Posts: 60647
Re: If [z] denotes the greatest integer less than or equal to z and [z] =  [#permalink]

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1
ammuseeru wrote:
Bunuel wrote:
sguptashared wrote:
If [z] denotes the greatest integer less than or equal to z and [z] = -1 which of the following statements must be true?

A. z = -1
B. -2 <= z < -1
C. -2 < z <= -1
D. -1 <= z < 0
E. -1 < z <= 0

[z] is the greatest integer less than or equal to z means that some function [] rounds DOWN a number to the nearest integer. For example [1.5]=1, =2, [-1.5]=-2, ...

Now, since [z] = -1, then $$-1\leq{z}<0$$ --> ANY number from this range when rounded down to the nearest integer is -1.

Similar questions to practice:
if-denotes-the-least-integer-greater-than-or-equal-to-x-132223.html
if-denotes-the-greatest-integer-less-than-or-equal-to-x-94687.html
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Hope it helps.

If question asks " [z] denotes the greatest integer less than or equal to z and [z] = 0", then can we say 0<=Z<1 ?

Regards,
Ammu

Yes. [z] = 0, would mean that $$0\leq{z}<1$$ --> ANY number from this range when rounded down to the nearest integer is 0.

Check other Rounding Functions Questions in our Special Questions Directory.

Hope it helps.
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Intern  Joined: 25 Dec 2012
Posts: 16
Re: If [z] denotes the greatest integer less than or equal to z and [z] =  [#permalink]

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1
Bunuel wrote:
sguptashared wrote:
If [z] denotes the greatest integer less than or equal to z and [z] = -1 which of the following statements must be true?

A. z = -1
B. -2 <= z < -1
C. -2 < z <= -1
D. -1 <= z < 0
E. -1 < z <= 0

[z] is the greatest integer less than or equal to z means that some function [] rounds DOWN a number to the nearest integer. For example [1.5]=1, =2, [-1.5]=-2, ...

Now, since [z] = -1, then $$-1\leq{z}<0$$ --> ANY number from this range when rounded down to the nearest integer is -1.

Similar questions to practice:
if-denotes-the-least-integer-greater-than-or-equal-to-x-132223.html
if-denotes-the-greatest-integer-less-than-or-equal-to-x-94687.html
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is-x-an-odd-integer-1-x-2-2x-99856.html
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is-the-greatest-integer-less-than-or-equal-to-the-real-n-154006.html

Hope it helps.

Hi Bunuel,

Just a question. Why do you think it could not be A?
Math Expert V
Joined: 02 Sep 2009
Posts: 60647
Re: If [z] denotes the greatest integer less than or equal to z and [z] =  [#permalink]

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2
gbascurs wrote:
Bunuel wrote:
sguptashared wrote:
If [z] denotes the greatest integer less than or equal to z and [z] = -1 which of the following statements must be true?

A. z = -1
B. -2 <= z < -1
C. -2 < z <= -1
D. -1 <= z < 0
E. -1 < z <= 0

[z] is the greatest integer less than or equal to z means that some function [] rounds DOWN a number to the nearest integer. For example [1.5]=1, =2, [-1.5]=-2, ...

Now, since [z] = -1, then $$-1\leq{z}<0$$ --> ANY number from this range when rounded down to the nearest integer is -1.

Similar questions to practice:
if-denotes-the-least-integer-greater-than-or-equal-to-x-132223.html
if-denotes-the-greatest-integer-less-than-or-equal-to-x-94687.html
for-all-numbers-t-let-t-be-defined-as-the-greatest-105416.html
is-x-an-odd-integer-1-x-2-2x-99856.html
for-all-z-denotes-the-least-integer-greater-than-or-110561.html
denotes-to-be-the-least-integer-no-less-than-x-is-130841.html
for-all-numbers-t-let-t-be-defined-as-the-greatest-105416.html
if-is-the-greatest-integer-less-than-or-equal-to-x-what-144277.html
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awkward-ds-105416.html
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if-denotes-the-greatest-integer-less-than-or-equal-to-x-94687.html
denotes-to-be-the-least-integer-no-less-than-x-is-130841.html
is-the-greatest-integer-less-than-or-equal-to-the-real-n-154006.html

Hope it helps.

Hi Bunuel,

Just a question. Why do you think it could not be A?

Because z is not necessarily -1, it could be any number from -1 to 0 ($$-1\leq{z}<0$$ ), for example, -1/2.

Check other Rounding Functions Questions in our Special Questions Directory.

Hope it helps.
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Re: If [z] denotes the greatest integer less than or equal to z and [z] =  [#permalink]

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But the question states that it is an integer... What do I missing?
Math Expert V
Joined: 02 Sep 2009
Posts: 60647
Re: If [z] denotes the greatest integer less than or equal to z and [z] =  [#permalink]

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gbascurs wrote:
But the question states that it is an integer... What do I missing?

The questions says that [z] is an integer, not z.

_________________
Intern  Joined: 25 Dec 2012
Posts: 16
Re: If [z] denotes the greatest integer less than or equal to z and [z] =  [#permalink]

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Bunuel wrote:
gbascurs wrote:
But the question states that it is an integer... What do I missing?

The questions says that [z] is an integer, not z.

Intern  Joined: 06 Aug 2014
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Schools: ISB '17 (A)
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Re: If [z] denotes the greatest integer less than or equal to z and [z] =  [#permalink]

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Dear Bunuel,

Although I completely agree that the answer is indeed option D here, however option A is not wrong also. If we plug in the value given in option A, we indeed satisfy the condition given.
Does the inclusion of option A not make choosing the correct answer slightly ambiguous. I mean why give an option such as A at all in the exam.
What trick am I missing here?

Math Expert V
Joined: 02 Sep 2009
Posts: 60647
Re: If [z] denotes the greatest integer less than or equal to z and [z] =  [#permalink]

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Dear Bunuel,

Although I completely agree that the answer is indeed option D here, however option A is not wrong also. If we plug in the value given in option A, we indeed satisfy the condition given.
Does the inclusion of option A not make choosing the correct answer slightly ambiguous. I mean why give an option such as A at all in the exam.
What trick am I missing here?

The question asks which of the following statements MUST be true, not could be true. z could be -1, but it could also be, for example, -1/2. Thus A is not a statement which must be true.

Check other Rounding Functions Questions in our Special Questions Directory.

Hope it helps.
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Joined: 28 Nov 2014
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Concentration: Strategy
Schools: Fisher '19 (M$) GPA: 3.71 Re: If [z] denotes the greatest integer less than or equal to z and [z] = [#permalink] ### Show Tags Bunuel wrote: madhavmarda wrote: Dear Bunuel, Although I completely agree that the answer is indeed option D here, however option A is not wrong also. If we plug in the value given in option A, we indeed satisfy the condition given. Does the inclusion of option A not make choosing the correct answer slightly ambiguous. I mean why give an option such as A at all in the exam. What trick am I missing here? Thanks in advance, Madhav The question asks which of the following statements MUST be true, not could be true. z could be -1, but it could also be, for example, -1/2. Thus A is not a statement which must be true. Check other Rounding Functions Questions in our Special Questions Directory. Hope it helps. I really cannot understand why A cannot be the answer. Z=-1 [Z] = -1 Condition satisfies. I have read the theory and solved all the questions in the link, I don't know what am I missing CEO  S Joined: 20 Mar 2014 Posts: 2548 Concentration: Finance, Strategy Schools: Kellogg '18 (M) GMAT 1: 750 Q49 V44 GPA: 3.7 WE: Engineering (Aerospace and Defense) Re: If [z] denotes the greatest integer less than or equal to z and [z] = [#permalink] ### Show Tags 2 anewbeginning wrote: Bunuel wrote: madhavmarda wrote: Dear Bunuel, Although I completely agree that the answer is indeed option D here, however option A is not wrong also. If we plug in the value given in option A, we indeed satisfy the condition given. Does the inclusion of option A not make choosing the correct answer slightly ambiguous. I mean why give an option such as A at all in the exam. What trick am I missing here? Thanks in advance, Madhav The question asks which of the following statements MUST be true, not could be true. z could be -1, but it could also be, for example, -1/2. Thus A is not a statement which must be true. Check other Rounding Functions Questions in our Special Questions Directory. Hope it helps. I really cannot understand why A cannot be the answer. Z=-1 [Z] = -1 Condition satisfies. I have read the theory and solved all the questions in the link, I don't know what am I missing The question is a MUST BE TRUE for ALL values. Not a could be true. Had this question been a 'could be true" question, then you would have stopped at option A itself and moved to the next question. But as this is a MUST BE TRUE question, you need to make sure that [z] =-1 is ONLY satisfied by z =-1 from the given options (and not by any other option). The correct answer to a MUST BE TRUE question, will nullify all other options. This is a very important point to remember. We are not denying that z = -1 is not true but what we are mentioning is that is it the ONLY possible choice ? Not necessarily z = -0.5 also satisfies the given condition. So if z = -0.5 also satisfies the given condition, then option D is true as well (along with option A, per your statement). A question can not have 2 correct answers. As I said before, in a MUST BE TRUE question, 1. Get the option(s) that satisfy a given condition 2. Eliminate all but 1 options. The remaining option will be the correct answer. There will always be concrete reasons to eliminate other options. If you are not able to eliminate options, then you are missing some important information. Unless you take care of both the options, you can not be sure of your answer. For this question, I adopted the following method: 1. Used z =-0.5 to eliminate options A-C 2. Used z = 0 to eliminate option E. If z =0 then [z] = 0 and $$\neq$$ -1 The only option remaining after the 2 steps above was D and is the correct answer. Current Student G Joined: 28 Nov 2014 Posts: 812 Concentration: Strategy Schools: Fisher '19 (M$)
GPA: 3.71
Re: If [z] denotes the greatest integer less than or equal to z and [z] =  [#permalink]

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Engr2012 wrote:
Dear Bunuel,

The question is a MUST BE TRUE for ALL values. Not a could be true. Had this question been a 'could be true" question, then you would have stopped at option A itself and moved to the next question.

But as this is a MUST BE TRUE question, you need to make sure that [z] =-1 is ONLY satisfied by z =-1 from the given options (and not by any other option). The correct answer to a MUST BE TRUE question, will nullify all other options. This is a very important point to remember. We are not denying that z = -1 is not true but what we are mentioning is that is it the ONLY possible choice ? Not necessarily z = -0.5 also satisfies the given condition. - This helped

So if z = -0.5 also satisfies the given condition, then option D is true as well (along with option A, per your statement). A question can not have 2 correct answers.

As I said before, in a MUST BE TRUE question,

1. Get the option(s) that satisfy a given condition
2. Eliminate all but 1 options. The remaining option will be the correct answer. There will always be concrete reasons to eliminate other options. If you are not able to eliminate options, then you are missing some important information.

Unless you take care of both the options, you can not be sure of your answer.

For this question, I adopted the following method:

1. Used z =-0.5 to eliminate options A-C
2. Used z = 0 to eliminate option E. If z =0 then [z] = 0 and $$\neq$$ -1

The only option remaining after the 2 steps above was D and is the correct answer.

Great explanation. I am fully satisfied with the explanation. Thanks a lot Originally posted by keats on 01 Aug 2015, 06:49.
Last edited by keats on 21 Aug 2016, 15:05, edited 2 times in total.
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Re: If [z] denotes the greatest integer less than or equal to z and [z] =  [#permalink]

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anewbeginning wrote:
Engr2012 wrote:
Dear Bunuel,

The question is a MUST BE TRUE for ALL values. Not a could be true. Had this question been a 'could be true" question, then you would have stopped at option A itself and moved to the next question.

But as this is a MUST BE TRUE question, you need to make sure that [z] =-1 is ONLY satisfied by z =-1 from the given options (and not by any other option). The correct answer to a MUST BE TRUE question, will nullify all other options. This is a very important point to remember. We are not denying that z = -1 is not true but what we are mentioning is that is it the ONLY possible choice ? Not necessarily z = -0.5 also satisfies the given condition. - This helped

So if z = -0.5 also satisfies the given condition, then option D is true as well (along with option A, per your statement). A question can not have 2 correct answers.

As I said before, in a MUST BE TRUE question,

1. Get the option(s) that satisfy a given condition
2. Eliminate all but 1 options. The remaining option will be the correct answer. There will always be concrete reasons to eliminate other options. If you are not able to eliminate options, then you are missing some important information.

Unless you take care of both the options, you can not be sure of your answer.

For this question, I adopted the following method:

1. Used z =-0.5 to eliminate options A-C
2. Used z = 0 to eliminate option E. If z =0 then [z] = 0 and $$\neq$$ -1

The only option remaining after the 2 steps above was D and is the correct answer.
[/quote]

Awesome explanation.

I am not satisfied with the explanation. Thanks a lot A lot of Kudos to you[/quote]

So are you or are you not satisfied with the explanation?
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Re: If [z] denotes the greatest integer less than or equal to z and [z] =  [#permalink]

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I'm sorry that was a TYPO Engr2012

I apologise for the same

Cheers
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Re: If [z] denotes the greatest integer less than or equal to z and [z] =  [#permalink]

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Bunuel wrote:
sguptashared wrote:
If [z] denotes the greatest integer less than or equal to z and [z] = -1 which of the following statements must be true?

A. z = -1
B. -2 <= z < -1
C. -2 < z <= -1
D. -1 <= z < 0
E. -1 < z <= 0

[z] is the greatest integer less than or equal to z means that some function [] rounds DOWN a number to the nearest integer. For example [1.5]=1, =2, [-1.5]=-2, ...

Now, since [z] = -1, then $$-1\leq{z}<0$$ --> ANY number from this range when rounded down to the nearest integer is -1.

Similar questions to practice:
if-denotes-the-least-integer-greater-than-or-equal-to-x-132223.html
if-denotes-the-greatest-integer-less-than-or-equal-to-x-94687.html
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is-x-an-odd-integer-1-x-2-2x-99856.html
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for-all-numbers-t-let-t-be-defined-as-the-greatest-105416.html
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if-denotes-the-greatest-integer-less-than-or-equal-to-x-94687.html
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is-the-greatest-integer-less-than-or-equal-to-the-real-n-154006.html

Hope it helps.

It helps as always!!
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Re: If [z] denotes the greatest integer less than or equal to z and [z] =  [#permalink]

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There is only 1 option which says -1<= Z , something that is directly mentioned in the question stem.
Hence option D
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Re: If [z] denotes the greatest integer less than or equal to z and [z] =  [#permalink]

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Top Contributor
sguptashared wrote:
If [z] denotes the greatest integer less than or equal to z and [z] = -1 which of the following statements must be true?

A. z = -1
B. -2 ≤ z < -1
C. -2 < z ≤ -1
D. -1 ≤ z < 0
E. -1 < z ≤ 0

Attachment:
prep.JPG

Let's make sure we understand what this [] symbol means.
Here are a few examples:
[3.1] = 3
[5.8] = 5
[0.7] = 0
[-0.9] = -1
[-4.6] = -5

So, if [z] = -1, then -1 ≤ z < 0

Cheers,
Brent
_________________ Re: If [z] denotes the greatest integer less than or equal to z and [z] =   [#permalink] 26 Feb 2019, 13:43
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