 Question

# A page is torn from a novel. The sum of the remaining page numbers is 10000. What are the two page-numbers on the torn page of this novel

CAT 2021

A
B
C
D
##### None of these
Solution

Correct option is

(D)

we know that sum of the consecutive integers from 1 to n is given by n(n + 1)/2. Let’s analyze each Roman numeral:

I: 77 and 78

If the page numbers on the torn page was 77 and 78, then the sum of the pages of the novel was 10,000 + 78 + 77 = 10,155. To determine whether 10,155 is the sum of consecutive integers, let’s solve n(n + 1)/2 = 10,155:

This equation has no integer roots. Thus, the page numbers on the torn page cannot be 77 and 78.

II: 33 and 34

If the page numbers on the torn page was 33 and 34, then the sum of the pages of the novel was 10,000 + 33 + 34 = 10,067. To determine whether 10,067 is the sum of consecutive integers, let’s solve n(n + 1)/2 = 10,067:

n(n+1)/2 = 10067

n^2 + n = 20134

n^2 + n - 20134 = 0

This equation has no integer roots. Thus, the page numbers on the torn page cannot be 33 and 34.

III: 8 and 9

If the page numbers on the torn page was 8 and 9, then the sum of the pages of the novel was 10,000 + 8 + 9 = 10,017. To determine whether 10,017 is the sum of consecutive integers, let’s solve n(n + 1)/2 = 10,017:

This equation has no integer roots. Thus, the page numbers on the torn page cannot be 8 and 9

hence , D (none of these ) is the correct option