# Technical Document Extraction: Mathematical Problem and Solution Attempts
## 1. Document Overview
This document presents a mathematical word problem regarding "proper divisors" and provides three distinct "Attempts" at solving it. The text uses a combination of standard black font for the problem statement and headers, and a monospaced font (alternating between green and red) for the step-by-step reasoning in the attempts.
## 2. Problem Statement (Header)
**Text:**
"The proper divisors of 12 are 1, 2, 3, 4 and 6. A proper divisor of an integer $N$ is a positive divisor of $N$ that is less than $N$. What is the sum of the proper divisors of the sum of the proper divisors of 284?"
---
## 3. Solution Attempts
### Attempt 1
This attempt follows a linear reasoning path but contains a calculation error regarding the definition of proper divisors for the second step.
* **Step 1:** Find proper divisors of 284: 1, 2, 4, 71, and 142.
* **Step 2:** Sum of proper divisors of 284: $1+2+4+71+142 = 219$.
* **Step 3:** Find proper divisors of 219: 1, 3, 73, and 219. *(Note: Including 219 contradicts the definition provided in the header).*
* **Step 4:** Sum of proper divisors of 219: $1+3+73+219 = 306$.
* **Conclusion:** The answer is 306.
* **Final Output:** `#### 306`
### Attempt 2
This attempt uses a conversational, repetitive dialogue style (alternating green and red lines) but reaches the same conclusion as Attempt 1.
* **Reasoning:**
* Identifies proper divisors of 284 as 1, 2, 4, 71, and 142.
* Calculates the sum as 219.
* Identifies proper divisors of 219 as 1, 3, 73, and 219. *(Note: Again, includes the number itself, violating the "less than N" rule).*
* Calculates the sum as $1+3+73+219 = 306$.
* **Conclusion:** The answer is 306.
* **Final Output:** `#### 306`
### Attempt 3
This attempt provides a different calculation for the initial sum and follows the "proper divisor" rule correctly for the second set.
* **Step 1:** Identifies proper divisors of 284 as 1, 2, 4, 71, and 142.
* **Step 2:** Sum of proper divisors of 284: $1+2+4+71+142 = 220$. *(Note: This differs from the 219 calculated in previous attempts).*
* **Step 3:** Find proper divisors of 220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110.
* **Step 4:** Sum of proper divisors of 220: $1+2+4+5+10+11+20+22+44+55+110 = 284$.
* **Conclusion:** The answer is 284.
* **Final Output:** `#### 284`
---
## 4. Technical Data Summary
| Component | Attempt 1 | Attempt 2 | Attempt 3 |
| :--- | :--- | :--- | :--- |
| **Divisors of 284** | 1, 2, 4, 71, 142 | 1, 2, 4, 71, 142 | 1, 2, 4, 71, 142 |
| **Sum of Divisors (284)** | 219 | 219 | 220 |
| **Divisors of Sum** | 1, 3, 73, 219 | 1, 3, 73, 219 | 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 |
| **Final Sum** | 306 | 306 | 284 |
| **Final Answer** | 306 | 306 | 284 |
## 5. Visual/Formatting Notes
* **Language:** English.
* **Color Coding:** The attempts use alternating green and red text lines, likely representing a multi-agent or "Chain of Thought" dialogue format.
* **Mathematical Notation:** Uses standard numerals and basic arithmetic operators (+, =). The variable $N$ is presented in LaTeX-style delimiters ($N$).
