Sam bought a 96page notebook and numbered the pages 1 throug

Sam bought a 96-page notebook and numbered the pages 1 through 192. He then tore out 25 pages and added the 50 numbers that he found on those pages. Could the number he got be 2016?

Solution

A 96-page notebook has 96 pages. Each page has two sides. Therefore, total number of available sides for writing is 2 x 96 = 192.

If the numbering of pages is considered as both sides of pages, then the pages will be numbered from 1 to 192.

Sam tore out 25 pages from the notebook and added the 50 numbers that he found on those pages.

There are several possibilities as stated below:

Case 1 – 25 consecutive pages

In this case, the numbers will form an arithmetic progression with common difference 1. If the first number is p and the last number is q, then the sum will be (p + q)*(50/2) = 25(p + q)

Therefore, 25(p + q) = 2016

Or, p + q = 2016/25

But, 2016 is not divisible by 25 and therefore 2016/25 is not an integer. This is a contradiction because (p + q) is an integer.

Hence, this case is not possible.

Case 2 – 25 non-consecutive pages

In this case, the numbers will not form an arithmetic progression. But if we look at both sides each page, there will be two consecutive numbers, one is even and another is odd. T herefore, the sum will be an odd number for each page (considering both sides).

The total sum will be sum of 25 odd numbers. 25 is an odd number.

(Even + Odd) + (Even + Odd) + …………… + (Even + Odd)

1^st Page               2^nd Page ……….. ………… 25^th Page

(Odd) + (Odd) + … + (Odd) = Odd

Because, odd sum of odd numbers is an odd number. Since, 2016 is an even number; this case is also not possible.

Therefore, the sum of the numbers obtained cannot be 2016. (Answer)