From the book, 25 pages were torn out. Can the sum of 50 numbers, which are the page numbers (on both sides) of these pages, be equal to 2001?

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The sum of two numbers written on both sides of one page gives a remainder of 3 when divided by 4.

## Solution

Consider one of the pages. On one side, an odd number $2 m-1$ is written, and on the other side, the next even number $2 m$. The sum of these two numbers is $4 m-1$, which gives a remainder of 3 when divided by 4. Therefore, the sum of 50 page numbers mentioned in the condition gives the same remainder when divided by 4 as $3 \cdot 25 = 75$, which is 3. However, 2001 gives a remainder of 1 when divided by 4.

## Answer

It cannot.