\n ## 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.