Question

Страницы книги пронумерованы подряд от первой до последней. Хулиган Вася вырвал из разных мест книги 25 листов и сложил номера всех 50 вырванных страниц. У него получилось число 2002. Когда об этом узнал Коля, он заявил, что при подсчёте Вася ошибся. Объясните, почему Коля действительно прав.

Answer 1

Номера страниц с двух сторон: на одной стороне четный номер, на другой - нечетный. получится 25 четных и 25 нечетных страниц. Сумма всех четных страниц дает четное число. Сумма двадцати пяти нечетных страниц дает нечетное число. Соответественно сумма всех страниц должна быть нечетной. У него получилось число 2002 - четное, следовательно Коля действительно прав .

Answer 2

Understanding the Problem

The problem states that the pages of a book are numbered consecutively from the first page to the last page. Vasya, a troublemaker, has torn out 25 pages from different places in the book and has arranged the numbers of all 50 torn-out pages. The sum of these page numbers is 2002. However, when Kolya found out about this, he claimed that Vasya made a mistake in his calculations. We need to explain why Kolya is correct.

Analyzing the Problem

To solve this problem, we need to find a contradiction in Vasya's claim that the sum of the page numbers of the torn-out pages is 2002. We know that Vasya tore out 25 pages, so the sum of the page numbers should be less than or equal to the sum of the first 25 positive integers, which is 325.

Finding the Contradiction

Let's assume that Vasya's claim is correct and the sum of the page numbers of the torn-out pages is indeed 2002. We can calculate the sum of the first 50 positive integers using the formula for the sum of an arithmetic series:

Sum = (n/2)(first term + last term)

In this case, the first term is 1 and the last term is 50, so the sum is:

Sum = (50/2)(1 + 50) = 25(51) = 1275

Since 2002 is greater than 1275, it is clear that Vasya's claim is incorrect. Therefore, Kolya is right in saying that Vasya made a mistake in his calculations.

Conclusion

In conclusion, Vasya made a mistake in his calculations when he claimed that the sum of the page numbers of the torn-out pages is 2002. The sum of the torn-out page numbers cannot exceed the sum of the first 25 positive integers, which is 325. Therefore, Kolya is correct in stating that Vasya made an error in his calculations.