Pages
Number Theory
Level
3
In a book there are 100 pages . From this book some pages are torn off. The sum of the pages remaining is 4949. How many pages are missing from the book?
The answer is 3.
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1 solution
Every page has two numbers on it. An odd number and that number +1.
Thus the sum of two consecutive numbers on a page is always odd with smaller number n the sum is 2n+1 odd.
The sum o f the remaining pages is 2 1 0 0 ∗ 1 0 1 − 4 9 = 1 0 1 an odd number. Since product of two odd numbers can only be odd, and the sum of numbers on a page is always odd, while the sum101 is also odd. So the number of pages must be odd.
Let us see. 2n+1 is an odd number with n as +tive ODD integer. So for one page. 2n+1 .. (2n +1) is not divide 101.
Next we try 3 pages. Average sum per page is 101/3 , 34.
Say a page with 15,16, sum 31. Left for two pages 101 - 31 = 70, one below and one above. Say we take 13,14= 27 and remaining 70 -27=43. (43-1)/2 = 21 and 22. Perfect!!
So we take out following 3 pages, (13,14),(15,16),(21,22)