# Technical Document Extraction: Proper Divisors Problem ## Problem Statement The proper divisors of an integer $N$ are positive divisors less than $N$. **Question**: What is the sum of the proper divisors of the sum of the proper divisors of 284? --- ## Attempt 1 - **Green Text**: "Ok, let's find the proper divisors of 284 first." "Right. So the proper divisors of 284 are 1, 2, 4, 71, and 142." "So the sum of the proper divisors of 284 is 1+2+4+71+142, which is 219." - **Red Text**: "Now let's find the proper divisors of 219." "The proper divisors of 219 are 1, 3, 73, and 219." "So the sum of the proper divisors of 219 is 1+3+73+219, which is 306." "So the answer to the question is 306." `#### 306` --- ## Attempt 2 - **Green Text**: "So the sum of the proper divisors of 284 is 1+2+4+71+142." "Ok." "And the proper divisors of 284 are 1,2,4,71, and 142." "Right." "And the sum of those proper divisors is 219." - **Red Text**: "So the sum of the proper divisors of 284 is the sum of the proper divisors of 219." "Right." "And the proper divisors of 219 are 1, 3, 73, and 219." "Right." "So the sum of the proper divisors of 219 is 1+3+73+219=306." "So the sum of the proper divisors of the sum of the proper divisors of 284 is 306." "Right." "So the answer to the question is 306." `#### 306` --- ## Attempt 3 - **Green Text**: "Right, so the proper divisors of 284 are 1, 2, 4, 71, and 142." "So the sum of the proper divisors of 284 is 1+2+4+71+142, which is 220." "Now let's find the proper divisors of 220." "The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110." "So the sum of the proper divisors of 220 is 1+2+4+5+10+11+20+22+44+55+110, which is 284." "So the answer to the question is 284." `#### 284` --- ## Analysis of Discrepancies 1. **Attempt 1 & 2**: - Incorrectly calculated the sum of proper divisors of 284 as **219** (actual sum is **220**). - Resulted in an incorrect final answer of **306**. 2. **Attempt 3**: - Correctly calculated the sum of proper divisors of 284 as **220**. - Proper divisors of 220 sum to **284**, making **284** the correct answer. --- ## Final Answer The sum of the proper divisors of the sum of the proper divisors of 284 is **284**. `#### 284`