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A botanist selects n^2 trees on an island and studies (2n + [#permalink]
06 Jun 2013, 19:20

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Question Stats:

25% (02:24) correct
75% (01:35) wrong based on 190 sessions

A botanist selects n^2 trees on an island and studies (2n + 1) trees everyday where n is an even integer. He does not study the same tree twice. Which of the following cannot be the number of trees that he studies on the last day of his exercise?

Re: A botanist select n^2 trees ... [#permalink]
06 Jun 2013, 20:22

First of all, two options(31,79) are very poorly framed & thus creating confusion. Anyways, the logic that the author wants to check is No of books that can be read on any day = 2n+1 , where n is even integer.

2n+1 = even + odd = odd So 2n+1 can never be even. Option C fits this bill. Hence the answer.

Note- botanist can not read 31, 79 books as well. But i am assuming there is sth wrong with these two options.

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Re: A botanist selects n^2 trees on an island and studies (2n + [#permalink]
06 Jun 2013, 23:54

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TirthankarP wrote:

A botanist selects n^2 trees on an island and studies (2n + 1) trees everyday where n is an even integer. He does not study the same tree twice. Which of the following cannot be the number of trees that he studies on the last day of his exercise?

A. 13 B. 28 C. 17 D. 31 E. 79

n=2 --> n^2=4 trees total--> 2n+1=5 trees studied everyday --> (last day)=4 (4 is the remainder when 4 is divided by 5); n=4 --> n^2=16 trees total --> 2n+1=9 trees studied everyday --> (last day)=7 (7 is the remainder when 16 is divided by 9); n=6 --> n^2=36 trees total --> 2n+1=13 trees studied everyday --> (last day)=10 (10 is the remainder when 36 is divided by 13); n=8 --> n^2=64 trees total --> 2n+1=17 trees studied everyday --> (last day)=13 (13 is the remainder when 64 is divided by 17); n=10 --> n^2=100 trees total --> 2n+1=21 trees studied everyday --> (last day)=16 (16 is the remainder when 100 is divided by 21); ...

(last day) = 4, 7, 10, 13, 16, ... a multiples of 3 plus 1. Only option C (17) does not fit.

Re: A botanist select n^2 trees ... [#permalink]
08 Sep 2013, 10:32

fameatop wrote:

First of all, two options(31,79) are very poorly framed & thus creating confusion. Anyways, the logic that the author wants to check is No of books that can be read on any day = 2n+1 , where n is even integer.

2n+1 = even + odd = odd So 2n+1 can never be even. Option C fits this bill. Hence the answer.

Note- botanist can not read 31, 79 books as well. But i am assuming there is sth wrong with these two options.

Fame

Hi fameatop

there is nothing wrong with the options.. think of it this way: the total no of trees to be studied is n^2, divisor is (2n+1).. now the question asks which of the following CANNOT be the remainder? (13, 28, 17, 31, 79)... if we just divide n^2 by (2n+1) quotient will be (n/2) and remainder will be (-n/2).. this remainder of (-n/2) can also be written as (2n + 1 - n/2) or (3n/2 + 1)... which means the remainder (or the no of trees on last day) will always be of the form (3n/2 + 1) where n is even (so 2 in denominator will be reduced/cancel out) which means this number will be of the form 3K + 1 where k is an integer... so whichever option does not satisfy this will be our answer... that is only one option, option C

(now i know some of you would be thinking 'how the hell is the quotient n/2 and remainder -n/2'... well that is a mathematical concept and i am not yet prepared to explain how it comes.... for you explanation by Bunuel is the best (anyday)..

Re: A botanist selects n^2 trees on an island and studies (2n + [#permalink]
16 Sep 2013, 01:59

Bunuel wrote:

TirthankarP wrote:

A botanist selects n^2 trees on an island and studies (2n + 1) trees everyday where n is an even integer. He does not study the same tree twice. Which of the following cannot be the number of trees that he studies on the last day of his exercise?

A. 13 B. 28 C. 17 D. 31 E. 79

n=2 --> n^2=4 trees total--> 2n+1=5 trees studied everyday --> (last day)=4 (4 is the remainder when 4 is divided by 5); n=4 --> n^2=16 trees total --> 2n+1=9 trees studied everyday --> (last day)=7 (7 is the remainder when 16 is divided by 9); n=6 --> n^2=36 trees total --> 2n+1=13 trees studied everyday --> (last day)=10 (10 is the remainder when 36 is divided by 13); n=8 --> n^2=64 trees total --> 2n+1=17 trees studied everyday --> (last day)=13 (13 is the remainder when 64 is divided by 17); n=10 --> n^2=100 trees total --> 2n+1=21 trees studied everyday --> (last day)=16 (16 is the remainder when 100 is divided by 21); ...

(last day) = 4, 7, 10, 13, 16, ... a multiples of 3 plus 1. Only option C (17) does not fit.

Answer: C.

Hope it's clear.

Is there any other approach or style to solve this question. _________________

Re: A botanist selects n^2 trees on an island and studies (2n + [#permalink]
09 Jan 2014, 06:59

TirthankarP wrote:

A botanist selects n^2 trees on an island and studies (2n + 1) trees everyday where n is an even integer. He does not study the same tree twice. Which of the following cannot be the number of trees that he studies on the last day of his exercise?

A. 13 B. 28 C. 17 D. 31 E. 79

There must be a more elegant way to solve this question than just plugging numbers and eliminating answer choices

We have that n^2 (2k^2) must of course be even while (2n+1) must be odd and a multiple of 4k + 1

We are basically asked for the remainder

When I divide 2k^2 / 4k + 1

I am then a bit stuck with the algebra cause I can't get rid of the 1 in the denominator to find possible remainders

Re: A botanist select n^2 trees ... [#permalink]
12 Jan 2014, 14:35

amanvermagmat wrote:

fameatop wrote:

First of all, two options(31,79) are very poorly framed & thus creating confusion. Anyways, the logic that the author wants to check is No of books that can be read on any day = 2n+1 , where n is even integer.

2n+1 = even + odd = odd So 2n+1 can never be even. Option C fits this bill. Hence the answer.

Note- botanist can not read 31, 79 books as well. But i am assuming there is sth wrong with these two options.

Fame

Hi fameatop

there is nothing wrong with the options.. think of it this way: the total no of trees to be studied is n^2, divisor is (2n+1).. now the question asks which of the following CANNOT be the remainder? (13, 28, 17, 31, 79)... if we just divide n^2 by (2n+1) quotient will be (n/2) and remainder will be (-n/2).. this remainder of (-n/2) can also be written as (2n + 1 - n/2) or (3n/2 + 1)... which means the remainder (or the no of trees on last day) will always be of the form (3n/2 + 1) where n is even (so 2 in denominator will be reduced/cancel out) which means this number will be of the form 3K + 1 where k is an integer... so whichever option does not satisfy this will be our answer... that is only one option, option C

(now i know some of you would be thinking 'how the hell is the quotient n/2 and remainder -n/2'... well that is a mathematical concept and i am not yet prepared to explain how it comes.... for you explanation by Bunuel is the best (anyday)..

Excuse me sir could you explain this part?

" if we just divide n^2 by (2n+1) quotient will be (n/2) and remainder will be (-n/2).. this remainder of (-n/2) can also be written as (2n + 1 - n/2) or (3n/2 + 1)... which means the remainder (or the no of trees on last day) will always be of the form (3n/2 + 1) where n is even (so 2 in denominator will be reduced/cancel out) which means this number will be of the form 3K + 1 where k is an integer"

Re: A botanist selects n^2 trees on an island and studies (2n + [#permalink]
14 Jan 2014, 00:49

Expert's post

HCalum11 wrote:

How can he ever study an even number of trees when n is an even integer? Won't (2n + 1) always be odd leaving B as the answer?

The question asks "which of the following cannot be the number of trees that he studies on the last day of his exercise?" So, even though 2n+1 is odd, last day there can be even number of trees left to study.

For example, if n=6, then there are total of n^2=36 trees and each day he studies 2n+1=13 trees. Thus on the first day he studies 13 trees, on the second day also 13 trees but on the last, 3rd day, there are only 36-13-13=10 trees left. Therefore on the last day he studies 10 trees.

Re: A botanist selects n^2 trees on an island and studies (2n + [#permalink]
14 Jan 2014, 03:17

n=2: There are n^2=4 trees in total Botanist studies (2n + 1)=5 trees everyday Last day=First day=4 trees remaining

n=4: There are n^2=16 trees in total Botanist studies 2n+1=9 trees everyday Last day=7

n=6: There are n^2=36 trees in total Botanist studies 2n+1=13 trees everyday Last day=36-(13trees x 2days)=10 (10 is the remainder when 36 is divided by 13); Note that (13trees x 3days) is bigger than 36

n=8: There are n^2=64 trees in total Botanist studies 2n+1=17 trees everyday Last day=64-(17trees x 3days)=13 (13 is the remainder when 64 is divided by 17); Note that (17trees x 4days) is bigger than 64

n=10: There are n^2=100 trees in total Botanist studies 2n+1=21 trees everyday Last day=100-(21trees x 4days)=16 (16 is the remainder when 100 is divided by 21); Note that (21trees x 5days) is bigger than 100

Re: A botanist selects n^2 trees on an island and studies (2n + [#permalink]
12 Mar 2014, 13:27

Bunuel wrote:

HCalum11 wrote:

How can he ever study an even number of trees when n is an even integer? Won't (2n + 1) always be odd leaving B as the answer?

The question asks "which of the following cannot be the number of trees that he studies on the last day of his exercise?" So, even though 2n+1 is odd, last day there can be even number of trees left to study.

For example, if n=6, then there are total of n^2=36 trees and each day he studies 2n+1=13 trees. Thus on the first day he studies 13 trees, on the second day also 13 trees but on the last, 3rd day, there are only 36-13-13=10 trees left. Therefore on the last day he studies 10 trees.

Hope it's clear.

Could one do something like the following?

n^2 = (2n+1) + r

n^2 - 2n +1 = r+2

(n-1)^2 = r+2 Now we are being asked about the remainder, so remainder would be a perfect square minus 2

But it doesn't seem to fit with the number choices

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