The largest 5-digit palindromic number that is divisible by 6 is 89898.
Understanding Palindromes and Divisibility by 6
Before identifying the specific number, let's understand what a palindrome and divisibility by 6 mean.
- Palindrome: A palindrome is a number (or word) that remains the same when its digits are reversed. Examples: 121, 1221, 45654.
- Divisibility by 6: A number is divisible by 6 if it's divisible by both 2 and 3.
- Divisibility by 2: The number must end in an even digit (0, 2, 4, 6, or 8).
- Divisibility by 3: The sum of its digits must be divisible by 3.
Finding the Largest Palindromic Number Divisible by 6
Given the reference, we know that 89898 is the largest 5-digit palindromic number divisible by 6. This information significantly helps in answering the question directly.
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5-Digit Palindrome Structure: A 5-digit palindrome has the form ABCBA, where A, B, and C are digits.
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Largest Value: To get the largest, we should aim for the largest digits for A first. So starting with 9, then a combination for B and C is checked.
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Divisibility Rules: The number must end in an even number, and the sum of its digits (2A+2B+C) should be divisible by 3.
In the case of 89898:
- It is a palindrome
- It ends in 8 which is an even number therefore divisible by 2.
- 8 + 9 + 8 + 9 + 8 = 42, which is divisible by 3, therefore, the number is divisible by 3.
- As 89898 is divisible by both 2 and 3, therefore it is divisible by 6.
Why not larger?
We know that the largest palindrome starting with '9' and followed by the largest digits cannot be divisible by 6.
For example, trying 99999 would fail because it’s not divisible by 2. And though 98989 is a palindrome, 9+8+9+8+9=43 which is not divisible by 3.
We can use this understanding of palindromes and divisibility by 6 to identify the largest 5-digit palindromic number that fulfills the criteria, which is 89898 as per our reference.
