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

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Question Stats: 51% (01:57) correct 49% (02:02) wrong based on 619 sessions

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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

Originally posted by atalpanditgmat on 08 Jun 2013, 01:54.
Last edited by Bunuel on 08 Jun 2013, 03:49, edited 2 times in total.
Edited the question.
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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.

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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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Haven't quite understood the Question;Can someone help what is it exactly asking?
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Re: [x] is the greatest integer less than or equal to the real n  [#permalink]

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

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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, [-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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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, [-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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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.

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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Posts: 62498
Re: [x] is the greatest integer less than or equal to the real n  [#permalink]

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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.

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, [-1.5]=-2, ...

17.6 rounded down to the nearest integer is also 17.

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

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

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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.

Hi Bunuel,

I was able to identify 289<n<=324, but I came with answer 36 because (324-289) + 1 = 36.

I think my mistake was I included 289 also in my calculation by adding 1. Whereas 289<n, so n should start from 290. Hence the OA is 35. Am I correct?
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Joined: 02 Sep 2009
Posts: 62498
Re: [x] is the greatest integer less than or equal to the real n  [#permalink]

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anupam87 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.

Hi Bunuel,

I was able to identify 289<n<=324, but I came with answer 36 because (324-289) + 1 = 36.

I think my mistake was I included 289 also in my calculation by adding 1. Whereas 289<n, so n should start from 290. Hence the OA is 35. Am I correct?

$$289\leq{n}<324$$. So, n can take all integer value from 289 to 323 inclusive: 323-289+1 = 35.
_________________ Re: [x] is the greatest integer less than or equal to the real n   [#permalink] 04 May 2019, 08:10
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