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e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3158
If [z] denotes the least integer greater  [#permalink]

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Difficulty:   85% (hard)

Question Stats: 46% (01:48) correct 54% (01:44) wrong based on 319 sessions

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Q.

If [z] denotes the least integer greater than or equal to z and [$$z^2$$] = 2, which of the following could be the value of [z]?

I. 2
II. 1
III. -2

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

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e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3158
Re: If [z] denotes the least integer greater  [#permalink]

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3
1
2

Official Solution

Given:

• The function [z]
• $$[z^2] = 2$$

To find: Can [z] be {2, 1, -2}?

Approach:

• To find which values of [z] are possible and which are not, we need to first know the range of possible values of z. Once we know what z can be, we’ll be able to find what [z] can be.

We’ll get an idea of the possible values of z from the fact that $$[z^2] = 2$$

Working Out:

• $$[z^2] = 2$$

• This means the least integer that is greater than or equal to $$z^2$$ is 2.

• We can write: $$1 < z^2 ≤ 2$$

• The above inequality contains 2 inequalities: $$z^2 > 1$$ AND $$z^2 ≤ 2$$
So now, we’ll solve these inequalities one by one, and then find the values of z that satisfy both these inequalities

• Solving $$z^2 > 1$$

• $$z^2 – 1 > 0$$

• $$(z+1)(z-1) > 0$$

This means, $$z < - 1$$ or $$z > 1$$ . . . (1)

• Solving $$z^2 ≤ 2$$

• $$z^2 – 2 ≤ 0$$

• $$(z + √2)(z-√2) ≤ 0$$

That is, $$-√2 ≤ z ≤ √2$$ . . . (2)

Combining (1) and (2):

Combining them to find their overlap zones and finding those values of z that satisfy both the inequalities

• So, either $$-√2 ≤ z < -1$$ or $$1 < z ≤ √2$$

• Either $$-√2 ≤ z < -1$$

• $$-1.4 ≤ z < -1$$

• In this case, $$[z] = -1$$

• Or $$1 < z ≤ √2$$

• $$1 < z ≤ 1.4$$

• $$[z] = 2$$
Thus, we see that [z] is either equal to -1 or equal to 2. Out of the 3 given values, only value I (which is 2) is therefore possible.

Looking at the answer choices, we see that the correct answer is Option A

Thanks,
Saquib
Quant Expert
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##### General Discussion
Senior Manager  G
Joined: 24 Apr 2016
Posts: 316
If [z] denotes the least integer greater  [#permalink]

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1
$$[z^2] = 2$$ means that

$$1<z^2\leq{2}$$

Now, Taking Square Root of both the sides.

If Z is positive, then

$$1<z\leq{\sqrt{2}}$$

then [z] = 2

If Z is negative, then

$$-1>z\geq{-\sqrt{2}}$$

then [z] = -1

Math Expert V
Joined: 02 Sep 2009
Posts: 59725
Re: If [z] denotes the least integer greater than or equal to z and [z2] =  [#permalink]

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1
2
niteshwaghray wrote:
If [z] denotes the least integer greater than or equal to z and [$$z^2$$] = 2, which of the following could be the value of [z]?

I) 2
II) 1
III) -2

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

Given function rounds up a number to the nearest integer. For example, [1.3] = 2, because 2 is the least integer greater than or equal to 1.3.

[$$z^2$$] = 2;

$$1 < z^2 \leq 2$$;

$$1 <|z| \leq \sqrt{2}$$ ($$\sqrt{2} \approx 1.4$$).

$$1 < z \leq \sqrt{2}$$ or $$-1 > z \geq \sqrt{2}$$

[z] can be 2 (if $$1 < z \leq \sqrt{2}$$) or -1 (if $$-1 > z \geq \sqrt{2}$$).

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Manager  P
Joined: 03 Mar 2018
Posts: 204
If [z] denotes the least integer greater than or equal to z and [z^2]  [#permalink]

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If [z] denotes the least integer greater than or equal to z and [z^2] = 2, which of the following could be the value of [z]?

I. 2
II. 1
III. -2

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) II and III only
_________________
Math Expert V
Joined: 02 Aug 2009
Posts: 8320
Re: If [z] denotes the least integer greater than or equal to z and [z^2]  [#permalink]

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2
1
itisSheldon wrote:
If [z] denotes the least integer greater than or equal to z and [z^2] = 2, which of the following could be the value of [z]?

I. 2
II. 1
III. -2

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) II and III only

Good question...
if [z^2] =2, $$z^2$$ lies between 2 and 3
therefore z lies between $$\sqrt{2}=1.4$$ and $$\sqrt{3}=1.7$$ or between $$-\sqrt{2}=-1.4$$ and $$-\sqrt{3}=-1.7$$ if z is negative..
so when z is positive it lies between 1.4 and 1.7 and the NEXT greater integer is 2 so [z]=2
but when z is negative it lies between -1.4 and -1.7 and the NEXT greater integer is -1 so [z]=-1.. Be careful here and do not take -2

Only 2 is given
ans A
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Math Expert V
Joined: 02 Sep 2009
Posts: 59725
Re: If [z] denotes the least integer greater  [#permalink]

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itisSheldon wrote:
If [z] denotes the least integer greater than or equal to z and [z^2] = 2, which of the following could be the value of [z]?

I. 2
II. 1
III. -2

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) II and III only

Merging topics. Please search before posting. Thank you.
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Intern  B
Joined: 07 Mar 2019
Posts: 28
Re: If [z] denotes the least integer greater  [#permalink]

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I don't understand, where does the z > 1 and z < -1 come from?
Intern  B
Joined: 24 Aug 2017
Posts: 21
GPA: 3.67
Re: If [z] denotes the least integer greater  [#permalink]

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Is my reasoning correct:

We know that [z] will take any number, (ex 1.4), and will round it up to the next highest integer, (2 in the case of 1.4) unless Z is already an integer.

Therefore, as others mentioned,

1<z^2 ≤ 2

Which means that √(Z^2), or Z is either larger than √1, which si 1 or -1, or smaller than √2 or -√2 (1.4, or -1.4)

-1 (or 1) <Z ≤ 1.4 ( or -1.4).

The different possibilities for [Z] are thus

1) 0, because if Z> -1 , the closest higher integer is 0,
2) 2, because if Z>1, Z < 1.4 , the closest integers are 2,
3) -1, because if Z> -1.4, the closest highest integer is -1.

The only option that appears in the answer choices is 2, thus the answer is A. Re: If [z] denotes the least integer greater   [#permalink] 03 Aug 2019, 13:20
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