The largest 5-digit palindromic number divisible by 6 is 89898.
Here's how we can determine that:
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Understanding Palindromes: A palindrome is a number (or word) that reads the same forwards and backward. A 5-digit palindrome has the form ABCBA, where A, B, and C are digits (0-9), and A cannot be 0.
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Divisibility Rule for 6: A number is divisible by 6 if it is divisible by both 2 and 3.
- Divisibility by 2: The number must be even (end in 0, 2, 4, 6, or 8).
- Divisibility by 3: The sum of the digits must be divisible by 3.
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Finding the Largest Palindrome: We want to start with the largest possible 5-digit palindrome, which would be something in the 90000s. We then work down ensuring the divisibility rules are met.
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Applying the Rules:
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To be divisible by 2, the last digit (and therefore the first digit, A) must be even. The largest even digit is 8. So our number is 8BCB8.
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To find the next largest palindrome, we want to maximize B. So, let's try B = 9: 89C98.
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Now for the divisibility rule of 3: 8 + 9 + C + 9 + 8 = 34 + C. We need 34 + C to be divisible by 3. The largest digit C can be to make this true is C = 8, since 34+8 = 42 is divisible by 3.
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This gives us the palindrome 89898.
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Verifying: 89898 / 6 = 14983. So 89898 is indeed divisible by 6.
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Therefore, 89898 is the largest 5-digit palindromic number divisible by 6.
