## Solution to: Missing Pages

Let the number of missing pages be *n* and the first missing page *p*+1.
Then the pages *p+1* up to and including *p+n* are missing, and *n* times the average of the numbers of the missing pages must be equal to 9808:

n×( ((p+1)+(p+n))/2 ) = 9808

In other words:

n×(2×p+n+1)/2 = 2×2×2×2×613

So:

n×(2×p+n+1) = 2×2×2×2×2×613

One of the two terms *n* and 2×*p*+*n*+1 must be even, and the other one must be odd.
Moreover, the term *n* must be smaller than the term 2×*p*+*n*+1.
It follows that there are only two solutions:

*n*=1 and 2×*p*+*n*+1=2×2×2×2×2×613, so*n*=1 and*p*=9808, so only page 9808 is missing.*n*=2×2×2×2×2 and 2×*p*+*n*+1=613, so*n*=32 and*p*=290, so the pages 291 up to and including 322 are missing.

Because it is asked which *pages* (plural) are missing, the solution is: the pages 291 up to and including 322 are missing.

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