Thread: Extremely interesting number theory problem?

1. Extremely interesting number theory problem?

A leaf is torn from a novel. The sum of the numbers on the remaining pages is 15000. What are the page numbers on the torn leaf?

2. Originally Posted by fardeen_gen
A leaf is torn from a novel. The sum of the numbers on the remaining pages is 15000. What are the page numbers on the torn leaf?
25 and 26

RonL

3. I know the answers.
(i found it out using the n(n+1)/2 formula and finding n for 15000 ,which yielded a non-whole value, and finding sum for the next greater whole value. The sum obtained i.e 15051 minus 15000 yielded 51 and since the torn leaf has consecutive page numbers, i got 25 and 26)

I would say the above method is extremely crude and wouldn't be accepted as a solution at all.Does anybody know the proper solution to this problem using number theory(as the question was listed in a number theory question bank)??

4. Originally Posted by fardeen_gen
I know the answers.
(i found it out using the n(n+1)/2 formula and finding n for 15000 ,which yielded a non-whole value, and finding sum for the next greater whole value. The sum obtained i.e 15051 minus 15000 yielded 51 and since the torn leaf has consecutive page numbers, i got 25 and 26)

I would say the above method is extremely crude and wouldn't be accepted as a solution at all.Does anybody know the proper solution to this problem using number theory(as the question was listed in a number theory question bank)??
It is a perfectly acceptable method. There are no rules in number theory about what methods are acceptable. You suppose that the number of pages is N, and then you bound N by considering the greatest and least total that the missing leaf would contribute, and you find only one integer that satisfies the constraints)

(not only does the removed leaf have one odd and one even number the odd number must be one less than the even.

5. Thank you Captain Black. Are there other methods also?

6. by the same logic

If n=174 ( no. of pages in the book) then the number on the torn leaf will be 112 and 113.