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[x] is the greatest integer less than or equal to the real n

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[x] is the greatest integer less than or equal to the real n [#permalink]

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[x] is the greatest integer less than or equal to the real number x. How many natural numbers n satisfy the equation \([\sqrt{n}] = 17\)?

A. 17
B. 34
C. 35
D. 36
E. 38
[Reveal] Spoiler: OA

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Last edited by Bunuel on 08 Jun 2013, 03:49, edited 2 times in total.
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Re: [x] is the greatest integer less than or equal to the real n [#permalink]

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atalpanditgmat wrote:
[x] is the greatest integer less than or equal to the real number x. How many natural numbers n satisfy the equation \([\sqrt{n}] = 17?\)

A. 17
B. 34
C. 35
D. 36
E. 38


From above we have that \(17\leq{\sqrt{n}}<18\) --> \(289\leq{n}<324\). Thus n can take 35 integer values from 289 to 323, inclusive.

Answer: C.
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Re: [x] is the greatest integer less than or equal to the real n [#permalink]

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New post 08 Jun 2013, 02:44
Haven't quite understood the Question;Can someone help what is it exactly asking?

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pavan2185 wrote:
Haven't quite understood the Question;Can someone help what is it exactly asking?


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

Now, since \([\sqrt{n}] = 17\), then \(17\leq{\sqrt{n}}<18\) --> ANy number from this range when rounded down to the nearest integer is 17.

Similar questions to practice:
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denotes-to-be-the-least-integer-no-less-than-x-is-130841.html

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Re: [x] is the greatest integer less than or equal to the real n [#permalink]

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New post 24 Aug 2013, 12:09
Bunuel wrote:
pavan2185 wrote:
Haven't quite understood the Question;Can someone help what is it exactly asking?


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

Now, since \([\sqrt{n}] = 17\), then \(17\leq{\sqrt{n}}<18\) --> ANy number from this range when rounded down to the nearest integer is 17.

Hope it helps.


We are only told that "[x] is the greatest integer less than or equal to the real number .."

And the answer is given as 1, so this makes 17<= sqrt(n) < 18

Where does the 18 come from? why cant it be 17<= sqrt (n) < 20??

Thanks

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Re: [x] is the greatest integer less than or equal to the real n [#permalink]

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New post 25 Aug 2013, 05:27
jjack0310 wrote:
Bunuel wrote:
pavan2185 wrote:
Haven't quite understood the Question;Can someone help what is it exactly asking?


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

Now, since \([\sqrt{n}] = 17\), then \(17\leq{\sqrt{n}}<18\) --> ANy number from this range when rounded down to the nearest integer is 17.

Hope it helps.


We are only told that "[x] is the greatest integer less than or equal to the real number .."

And the answer is given as 1, so this makes 17<= sqrt(n) < 18

Where does the 18 come from? why cant it be 17<= sqrt (n) < 20??

Thanks


\([x] = 17\) means that \(17\leq{x}<18\), because ANY number from this range rounded down to the nearest integer is 17. It cannot be \(17\leq{x}<20\), because if x is for example 19,5, then [19.5]=19 and not 17.

Hope it's clear.
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Re: [x] is the greatest integer less than or equal to the real n [#permalink]

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New post 12 Nov 2013, 04:44
Bunuel wrote:
atalpanditgmat wrote:
[x] is the greatest integer less than or equal to the real number x. How many natural numbers n satisfy the equation \([\sqrt{n}] = 17?\)

A. 17
B. 34
C. 35
D. 36
E. 38


From above we have that \(17\leq{\sqrt{n}}<18\) --> \(289\leq{n}<324\). Thus n can take 35 integer values from 289 to 323, inclusive.

Answer: C.

Bunuel,

Shouldn't it be upto 17.5 then because you could round it off to 18 if it was 17.6?

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Re: [x] is the greatest integer less than or equal to the real n [#permalink]

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New post 12 Nov 2013, 06:19
RahulAA wrote:
Bunuel wrote:
atalpanditgmat wrote:
[x] is the greatest integer less than or equal to the real number x. How many natural numbers n satisfy the equation \([\sqrt{n}] = 17?\)

A. 17
B. 34
C. 35
D. 36
E. 38


From above we have that \(17\leq{\sqrt{n}}<18\) --> \(289\leq{n}<324\). Thus n can take 35 integer values from 289 to 323, inclusive.

Answer: C.

Bunuel,

Shouldn't it be upto 17.5 then because you could round it off to 18 if it was 17.6?


No. "[x] is the greatest integer less than or equal to the real number x" means that some function [] rounds DOWN a number to the nearest integer. For example [1.5]=1, [2]=2, [-1.5]=-2, ...

17.6 rounded down to the nearest integer is also 17.

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
how-x-become-fraction-if-its-been-said-an-integer-94687.html
awkward-ds-105416.html
help-data-sufficiency-equation-and-inequalities-99856.html
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
if-denotes-the-greatest-integer-less-than-or-equal-to-z-162078.html
if-denotes-the-least-integer-greater-than-or-equal-to-x-132223.html

Hope it helps.
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PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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[x] is the greatest integer less than or equal to the real n [#permalink]

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Bunuel, just a quick shortcut to get this answer very quickly.

We want \(17 \leq{sqrt n} <{18}\). Therefore, we need to know how many numbers are between \(17^2\) and \(18^2.\).

\(18^2 - 17^2 = (18+17)(18-17) = 35.\).

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Re: [x] is the greatest integer less than or equal to the real n [#permalink]

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New post 07 Aug 2017, 10:35
I had a slightly different approach. the Q says [x] is GREATEST integer is less than equal to x --> so it is approaching x value and we are to consider the lower values of any decimal/ non integer value given as the value of x.
now { sq rt n} = 17
we know sq root 16 = 4; so root of 17 is about 4.5 or little more than 4 but not 5 (approaching 5)
so we get 1 -2 (10 values in decimals); 2 - 3 (10 more values); 3 - 4 (10 more) and 4 - 5 (10 more) -- I did this because the it is approaching x and we are to consider values which are lesser than x.
Total 40 values --> 40 - 5 (5 for repeat/ overlap) = 35 values
Would this reasoning be correct?
I feel Bunuel's method was much more succinct though!

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Re: [x] is the greatest integer less than or equal to the real n   [#permalink] 07 Aug 2017, 10:35
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