Nina bought a notebook containing 96 pages and numbered on both sides from 1 to 192.Sumit tore out 20 pages of Nina's notebook And added page numbers he found on this torn pages.Is it possible that sumit Got 2023as the sum?Give reasons to your answer.(

Let's break this problem down logically by looking at the structure of the notebook and the sum of the page numbers on the torn pages.

### Structure of the Notebook:
- The notebook contains 96 pages.
- Each page is numbered on both sides, so there are 192 total page numbers in the notebook.
- The numbering pattern goes as follows: Page 1 has numbers 1 and 2, Page 2 has numbers 3 and 4, and so on, until Page 96, which has numbers 191 and 192.

Each sheet (a piece of paper) in the notebook consists of **two pages** with four numbers:
- For example, the first sheet has page numbers 1, 2, 191, and 192.

### Total Sum of All Page Numbers:
The sum of all the numbers from 1 to 192 is the sum of an arithmetic series:
\[
\text{Sum} = \frac{192 \times (192 + 1)}{2} = \frac{192 \times 193}{2} = 18528
\]

Thus, the total sum of all page numbers in the notebook is **18,528**.

### Sumit Tears Out 20 Pages:
- Each "page" that is torn out refers to a single leaf of paper containing 2 page numbers, one on each side.
- Since Sumit tore out 20 pages, the total number of page numbers torn out would be **40 page numbers** (20 sheets with 2 page numbers each).

### Can the Sum of Torn Pages Equal 2023?
Let’s evaluate if it’s possible for the sum of the numbers on these 40 torn-out pages to equal 2023.

1. **Parity Check**:
   - The total sum of page numbers from 1 to 192 is **18,528**, which is even.
   - If we subtract a subset of page numbers (the numbers on the torn pages), their sum must be even to result in an even total.
   - 2023 is **odd**. If Sumit adds up the numbers on the torn pages and gets 2023, the difference between 18,528 and this sum would need to be even for the total sum to still be 18,528.
   - However, 18,528 minus 2023 results in **16,505**, which is odd. This creates a contradiction because the total sum of the remaining page numbers must also be even.

### Conclusion:
It is **not possible** for Sumit to have gotten 2023 as the sum of the torn page numbers. The sum of any subset of numbers from 1 to 192 must always be even, given that the total sum of all page numbers is even, but 2023 is odd. Therefore, the sum of 2023 cannot occur.

### Do you want more details or have any questions?
Here are some related questions for further exploration:
1. What would be the total sum if Sumit tore out a different number of pages?
2. How could you determine the sum of all remaining pages if Sumit tore out a different set of pages?
3. What would be the effect of tearing out an odd number of pages instead of 20?
4. How does the even/odd property of numbers influence the sum in this kind of problem?
5. What other methods can be used to prove that 2023 is not possible?

**Tip:** Always check the parity (even/odd nature) of the total sum and the numbers involved when solving problems involving sums and differences.

Explore whether Sumit can achieve 2023 as the sum of page numbers torn from Nina's notebook. Understand the structure of the notebook, the total sum of all page numbers, and why the sum of 2023 is not possible. Suitable for Grades 7-9, this problem delves into arithmetic series, even and odd numbers, and basic algebraic reasoning.

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