A child tore off one leaf from a book having page numbers from 1 to N. Summation of the remaining number of pages = 605. Which are the page numbers on that leaf torn off this book?
Answers
Given : A child tore off one leaf from a book having page numbers from 1 to N. Summation of the remaining number of pages = 605
To Find : page numbers on that leaf torn off this book
Solution:
page numbers on that leaf torn off this book = a , a + 1
1 ≤ a , a + 1 ≤ N
page numbers from 1 to N.
sum = N (N + 1)/2
N (N + 1)/2 = 605 + a + a + 1
=> N (N + 1)/2 = 606 + 2a
=> N (N + 1) = 1212 + 4a
=> N(N + 1) > 1212
34 * 35 = 1190 < 1212
35 * 36 = 1260
1260 = 1212 + 4a
=> a = 12
12 and 13 page number
36 * 37 = 1332
1332 = 1212 + 4a
=> a= 30
Hence page numbers are 30 and 31
37 * 38 = 1406
1406 = 1212 + 4a
=> a= 48.5 > 37 hence not possible
if N = 35 then the page numbers on that leaf torn off this book are 12 and 13
if N = 36 then the page numbers on that leaf torn off this book are 30 and 31
As page number starts with 1
so smaller number on leaf should be odd and larger even
so 30 and 31 is possible
page numbers on that leaf torn off this book 30 and 31
total page numbers = 36
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