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

Question

is the greatest integer less than or equal to the real number x. How many natural numbers n satisfy the equation    = 17 ?

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

Bunuel · Accepted Answer

From above we have that  17\leq{\sqrt{n}}<18

Square:  289\leq{n}<324 . Thus n can take 35 integer values from 289 to 323, inclusive.

Answer: C.

GSBae · Answer

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. .

pavan2185 · Answer

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

Bunuel · Answer

" 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, =2, =-2, ... Now, since = 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: 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 Hope it helps.

jjack0310 · Answer

We are only told that "  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

Bunuel · Answer

= 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 and not 17.

Hope it's clear.

RahulAA · Answer

Bunuel,

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

Bunuel · Answer

No. " 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, =2, =-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.

Madhavi1990 · Answer

I had a slightly different approach. the Q says   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!

anupam87 · Answer

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?

Bunuel · Answer

289\leq{n}<324 . So, n can take all integer value from 289 to 323 inclusive: 323-289+1 = 35.

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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

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