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just one page (number 9808) is missing ##### Share on other sites

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If an even number of pages are removed the sum of the missing page numbers  divided by the number of pages missing would have a fractional portion of 0.5

If an odd number of pages are removed the sum of the missing page numbers  divided by the number of pages missing would have a fractional portion of 0.

A trivial solution exist for the removal of a single page [number 9808]. When divided by 1, the result is 9808.

If we ignore this [the problem states that "pages" are missing].

If we define N as the number of pages missing then we must test the possible number of missing pages using the following:

Frac [9808/ N] =

• 0.5 where N is an even number; or
• 0.0 where N is an off number

testing for up to 100 pages missing there is only 1 value of N that satisfies this criteria

For N = 32

9808/ 32 = 306.5

This tells us that 32 pages are missing and the average value of these page numbers is 306.5

Taking 16 integers values below and 16 integer values above 306.5 gives us a range of

pages 291 thru 322

The sum of these pages = 9808

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Let p be the first page and let n be the number of pages.

We have to solve the following equation:
n*(2p+n-1)/2 = 9808 = 613*16
or equivalent one:
n*(2p+n-1) = 613*2^5
in positive integers (note that 613 is prime).
If n is a multiple of 613, then left side is obviously bigger than right side,
so n has to be equal to 2^k for some 0<=k<=5.
In such case we can divide both sides by n,
and see that left side becomes odd and right side stays even, unless n=1 or n=32.
This leads to two mentioned above solutions.
Edited by witzar
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Only one of these two solutions actually make sense, though, because a page in a book has two sides, and therefore two page numbers.

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unless page 9808 was the last numbered page.

But the reason why i prefer the other solution is that the op requested consecutive pages.

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