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## Textual Document: Problem Solving Steps
### Overview
The image presents a step-by-step reasoning process for solving a mathematical problem. The problem asks to determine the least positive integer *x* such that the sum of *x* and 4609 gives a remainder of 2104 when divided by 12. The document outlines initial reasoning steps, a chosen step, and a rejected step.
### Components/Axes
The document is structured into sections labeled: "prompt", "initial_reasoning_steps", "chosen_step", and "rejected_step". There are no axes or charts present.
### Content Details
**prompt:**
"Determine the least positive integer x for which the sum of x and 4609 gives a remainder of 2104 when divided by 12."
**initial_reasoning_steps:**
"Let's think step by step.
Step 1: We know that x + 4609 ≡ 2104 (mod 12).
Step 2: We can rewrite the given congruence as x ≡ 2104 - 4609 (mod 12).
Step 3: Calculating 2104 - 4609 = -2505.
Step 4:"
**chosen_step:**
"Now we need to find the equivalent positive integer for -2505 (mod 12).
To do this, we divide -2505 by 12 and find the remainder. -2505 ÷ 12 = -208 remainder -9.
However, we want a positive remainder, so we add 12 to -9 to get a positive remainder. -9+12=3.
So, -2505 ≡ 3 (mod 12)."
**rejected_step:**
"We want to find the positive integer x, so we add 12until we get a positive number. -2505 + 12 = -210 + 12 = 30."
### Key Observations
The document demonstrates a modular arithmetic problem-solving approach. The "chosen_step" correctly identifies the positive remainder when -2505 is divided by 12. The "rejected_step" attempts to find a positive integer by repeatedly adding 12, but the calculation is incomplete and leads to an incorrect result.
### Interpretation
The document illustrates a common problem-solving technique: breaking down a complex problem into smaller, manageable steps. The inclusion of both a "chosen_step" and a "rejected_step" suggests an iterative process of reasoning and refinement. The "chosen_step" demonstrates the correct application of modular arithmetic to find the equivalent positive integer, while the "rejected_step" highlights a potential pitfall in the reasoning process. The document is a record of thought, and not a final solution. The final answer is not provided.