A page is torn from a novel. The sum of the remaining page numbers is GMAT Problem Solving

Question: A page is torn from a novel. The sum of the remaining page numbers is 10,000. What could be the page-numbers on the torn page?

I) 76 and 77
II) 33 and 34
III) and 6

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

Answer: E

Approach Solution (1):
n (n + 1)/n should be nearby 10,000
So n (n + 1) is somewhere near 20,000
So n should be around  about 141
142 * 143/2 = 10153
141 * 142/2 = 10011
So, the missing pages are either 76 and 77 or 5 and 6
Correct option: E

Approach Solution (2):
Sum of page numbers of n pages =n(n+1)/2
This should be greater than 10000
Hence minimum value of n = 141
141 x 142/2 = 10011 = 10000 + 11
11 = 5 + 6
5 + 6 = 11 or 76 + 77 = 153
Correct option: E

Approach Solution (3):
Here the sum should be close to 10,000 and greater than 10,000
Lets calculate (1/2) * n(n + 1) = 10,000
n * (n + 1) = 20,000
n = 141 approx
141 * 142 = 20022
(1/2) * 141 * 142 = 10011
So we are short of 11
Or
We are short of 142 (next page) + 11 = 153
1). 76 and 77 Sum of 76 and 77 = 153 possible
2). 33 and 34
3). 5 and 6. Sum of 5 and 6 = 11 possible
Correct option: E

“A page is torn from a novel. The sum of the remaining page numbers is 10,000. What could be the page-numbers on the torn page?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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