{
  "problem_id": "math_2",
  "domain": "math",
  "difficulty": "hard",
  "prompt": "Prove that for any positive integer n, the number n^3 - n is always divisible by 6.",
  "num_total_tokens": 2807,
  "num_thinking_tokens_approx": 2277,
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  "thinking_text": "\nOkay, so I need to prove that for any positive integer n, the number n\u00b3 - n is divisible by 6. Hmm, let me think about how to approach this. Divisibility by 6 means that the number must be divisible by both 2 and 3, right? So maybe I can show that n\u00b3 - n is divisible by 2 and by 3 separately. That seems like a good plan. Let me start by factoring the expression n\u00b3 - n.\n\nFirst, factor out an n: n\u00b3 - n = n(n\u00b2 - 1). Then, I notice that n\u00b2 - 1 is a difference of squares, so it factors further into (n - 1)(n + 1). So altogether, n\u00b3 - n factors into n(n - 1)(n + 1). \n\nSo n\u00b3 - n = (n - 1)n(n + 1). That's three consecutive integers multiplied together: n - 1, n, and n + 1. Wait, because if n is an integer, then those three numbers are consecutive. For example, if n = 2, then it's 1, 2, 3; if n = 5, it's 4, 5, 6, etc. So, three consecutive integers. \n\nNow, I remember that in any set of three consecutive integers, there must be at least one multiple of 2 and at least one multiple of 3. Therefore, their product must be divisible by both 2 and 3, and hence by 6. But maybe I should elaborate more on why exactly that is true. Let me think.\n\nStarting with divisibility by 2: Among any two consecutive integers, one is even. So in three consecutive integers, there are at least two even numbers? Wait, no. For example, take 1, 2, 3: only 2 is even. Take 2, 3, 4: 2 and 4 are even. So actually, in three consecutive numbers, there is at least one even number, but sometimes two. But regardless, since there is at least one, the product will be divisible by 2. Similarly, for divisibility by 3: In any three consecutive integers, one of them must be a multiple of 3. Because every third number is divisible by 3. So, if you have three in a row, one has to be. \n\nTherefore, since the product (n - 1)n(n + 1) is divisible by both 2 and 3, it must be divisible by 6. Therefore, n\u00b3 - n is divisible by 6. \n\nBut maybe I should formalize this more. Let me try to write it as a proof.\n\nFirst, factor n\u00b3 - n",
  "response_text": "\n\nTo prove that for any positive integer $ n $, the expression $ n^3 - n $ is divisible by 6, we can approach this in a clear and structured way:\n\n---\n\n### **Step 1: Factor the Expression**\n\nWe begin by factoring the expression:\n\n$$\nn^3 - n = n(n^2 - 1) = n(n - 1)(n + 1)\n$$\n\nThis is a product of **three consecutive integers**: $ (n - 1), n, (n + 1) $.\n\n---\n\n### **Step 2: Divisibility by 2**\n\nIn any set of three consecutive integers, at least **one is even**. This is because in any pair of consecutive integers, one is even. So, among three consecutive integers, we are guaranteed at least one even number. Therefore, the product:\n\n$$\n(n - 1)n(n + 1)\n$$\n\nis divisible by 2.\n\n---\n\n### **Step 3: Divisibility by 3**\n\nSimilarly, in any set of three consecutive integers, one of them is divisible by 3. This is because the sequence of integers modulo 3 cycles every three numbers: 0, 1, 2. Thus, any three consecutive numbers must cover all three residues modulo 3, and one of them must be a multiple of 3.\n\nHence, the product is divisible by 3.\n\n---\n\n### **Step 4: Combining the Two**\n\nSince the product $ (n - 1)n(n + 1) $ is divisible by both 2 and 3, and since 2 and 3 are coprime, it follows that the product is divisible by:\n\n$$\n2 \\times 3 = 6\n$$\n\nTherefore:\n\n$$\nn^3 - n = (n - 1)n(n + 1)\n$$\n\nis divisible by 6 for all positive integers $ n $.\n\n---\n\n### **Alternative Methods (Verification)**\n\n- **Modular Arithmetic**: We can test all residues modulo 6 and verify that $ n^3 - n \\equiv 0 \\mod 6 $ in all cases.\n- **Mathematical Induction**: We can use induction to confirm that if $ k^3 - k $ is divisible by 6, then so is $ (k + 1)^3 - (k + 1) $, with base case $ n = 1 $.\n\nAll approaches confirm the same result.\n\n---\n\n### **Final Answer**\n\n$$\n\\boxed{6}\n$$<|im_end|>",
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