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Director  P
Joined: 20 Jul 2017
Posts: 637
Location: India
Concentration: Entrepreneurship, Marketing
WE: Education (Education)
The sum of the page numbers after a page is removed from a  [#permalink]

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4 00:00

Difficulty:   75% (hard)

Question Stats: 48% (03:44) correct 52% (02:20) wrong based on 29 sessions

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The sum of the page numbers after a page is removed from a book is 1000. Find the 2 page numbers on the removed page of this book.

A. 7 and 8
B. 10 and 11
C. 17 and 18
D. 21 and 22
E. 5 and 6

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Intern  B
Joined: 02 Mar 2019
Posts: 5
Re: The sum of the page numbers after a page is removed from a  [#permalink]

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Sum of 45 number 1035
So sum of missing page number 35
Option C

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Director  P
Joined: 19 Oct 2018
Posts: 791
Location: India
Re: The sum of the page numbers after a page is removed from a  [#permalink]

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Let total number of pages in the book=n
Page numbers of torn pages are x and x+1, where x<n.

$$\frac{n*(n+1)}{2} -x-(x+1)=1000$$
$$n^2+n-4x-2002=0$$

As n has integral values, hence $$b^2-4ac$$ must be a perfect square.
$$b^2-4ac$$= 1-4(-4x-2002)=16x+8009

Nearest perfect square to 8009 is $$90^2$$, but their difference is not a multiple of 16. Next nearest perfect square is $$91^2$$, and difference between 91^2 and 8009 is equal to 272, which is a multiple of 16.
Hence x=272/16=17

C

Dillesh4096 wrote:
The sum of the page numbers after a page is removed from a book is 1000. Find the 2 page numbers on the removed page of this book.

A. 7 and 8
B. 10 and 11
C. 17 and 18
D. 21 and 22
E. 5 and 6

Posted from my mobile device
Intern  B
Joined: 11 Jun 2019
Posts: 23
Location: India
The sum of the page numbers after a page is removed from a  [#permalink]

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1
Sum of n numbers in arithmetic progression is N x [2A + D(n-1)]/2, where
N= Number of terms....(we dont know this from in given problem)
A= 1st term (has to be 1 as its the 1st page of the book)
D= constant difference (1 as numbers must be in sequence

After simplifying above formula, we get sum of terms = N/2.(N+1).
Now we know the sum of page numbers without missing pages is 1000. Hence sum of terms must be 1000 + sum of missing number.
Now we plug the values in given options we can conclude that after only option C yields integer value of N. Hence C
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Re: The sum of the page numbers after a page is removed from a  [#permalink]

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Dillesh4096 wrote:
The sum of the page numbers after a page is removed from a book is 1000. Find the 2 page numbers on the removed page of this book.

A. 7 and 8
B. 10 and 11
C. 17 and 18
D. 21 and 22
E. 5 and 6

Posted from my mobile device

Looking at the answer choices, we notice that the sum of page numbers removed from the book could be as small as 5 + 6 = 11 or as large as 21 + 22 = 43. Therefore, the actual sum of the pages before any pages were removed should be between 1000 + 11 = 1011 and 1000 + 43 = 1043.

Let’s test some numbers for the pages of books until we find a number between 1011 and 1043. As the formula for the sum of consecutive integers from 1 to n is n(n + 1)/2 and the square root of 1000 is around 30, a good place to start is:

(30 * 31)/2 = 465 - too low

(50 * 51)/2 = 1275 - too high

(40 * 41)/2 = 820 - too low

(45 * 46)/2 = 1035 - success

Since n = 45 results in a number between 1011 and 1043, the book can have a total of 45 pages. All we have to do now is to find an answer choice which adds up to 1035 - 1000 = 35.

A) 7 + 8 = 15

B) 10 + 11 = 21

C) 17 + 18 = 35

So, we know that the book had 45 pages and the sum of the page numbers was 1035 before any page was removed. After the page with page numbers 17 and 18 was removed, the sum of the remaining numbers was 1035 - (17 + 18) = 1000. The answer is C.

Answer: C
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