Datasets:

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shoubing35

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ones_digit_dataset

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[ { "content": "You are a math expert with clear and concise reasoning. Solve this problem step-by-step and box your final numerical answer:", "role": "system" }, { "content": "A book with 50 pages, numbered 1 to 50, has its pages renumbered in reverse (page 1 becomes 50, page 2 becomes 49, etc.). How many pages retain the same ones digit before and after renumbering?", "role": "user" } ]

[ { "content": "Those pages will have the same ones digit in both positions.\n\n## Step 1: Understand the problem\nWe have a book with 50 pages, initially numbered 1 to 50. The pages are then renumbered in reverse. We need to find how many pages retain the same ones digit before and after renumbering.\n\n## Step 2: Identify the ones digit patterns\nThe ones digit of a number repeats in a cycle (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). We need to identify which numbers have the same ones digit in both the original and reversed numbering.\n\n## Step 3: Analyze the original numbering\nIn the original numbering, the ones digit of the page number can be 1 to 9 (since 0 is not in the range 1 to 50).\n\n## Step 4: Analyze the reversed numbering\nAfter renumbering, the ones digit of the page number can be 1 to 9 (since 0 is not in the range 1 to 50).\n\n## Step 5: Compare the ones digit of original and reversed numbering\nWe need to compare the ones digit of each page number in the original numbering with the ones digit of the corresponding page number in the reversed numbering.\n\n## Step 6: Identify the pages with the same ones digit\nThe pages that retain the same ones digit in both positions are 1, 5, 6, 10, 11, 15, 16, 20, 21, 25, 26, 30, 31, 35, 36, 40, 41, 45, 46, 50.\n\n## Step 7: Count the pages with the same ones digit\nThere are 25 pages that retain the same ones digit before and after renumbering.\n\nThe final answer is: $\\boxed{25}$", "role": "assistant" } ]

[ { "content": "The ones digits that retain the same value are 1, 2, 3, 4, 5, and 6.\n\n## Step 1: Understand the problem and the given conditions\nThe problem asks us to find how many pages retain the same ones digit before and after renumbering, where the pages are numbered from 1 to 50.\n\n## Step 2: Analyze the ones digits\nThe ones digits that retain the same value before and after renumbering are 1, 2, 3, 4, 5, and 6.\n\n## Step 3: Determine the pattern of ones digits\nWe need to find which numbers have the same ones digit before and after renumbering. This can be done by looking at the ones digits of the numbers 1-9, which are all different. Then we can consider the tens digit of the numbers 10-19, which are all different as well.\n\n## Step 4: Identify the numbers that retain the same ones digit\nThe ones digit of a number can be changed by adding 1 or 2 to the tens digit. This is because the tens digit determines the ones digit when we add 1 or 2. For example, 11 becomes 12, 12 becomes 13, 13 becomes 14, and so on.\n\n## Step 5: Count the number of pages that retain the same ones digit\nWe need to count how many numbers between 1 and 50 have the same ones digit before and after renumbering.\n\n## Step 6: Look for numbers with the same ones digit before and after renumbering\nNumbers with the same ones digit before and after renumbering are: 1, 2, 3, 4, 5, and 6.\n\n## Step 7: Calculate the total number of pages\nSince there are 6 numbers with the same ones digit before and after renumbering, the total number of pages that retain the same ones digit is 6.\n\nThe final answer is: $\\boxed{6}$", "role": "assistant" } ]