The number of 7-digit palindromes divisible by 3 is infinite.
Understanding Palindromes and Divisibility by 3
A 7-digit palindrome reads the same forwards and backwards. This means it takes the form ABCDCBA, where A, B, C, and D are digits from 0-9. However, A cannot be 0, since we need a 7-digit number.
A number is divisible by 3 if the sum of its digits is divisible by 3. For our palindrome ABCDCBA, the sum of digits is 2A + 2B + 2C + D. This can be rewritten as 2(A + B + C) + D.
Infinite Possibilities
The given reference states, "The total number of 7-digit palindromes divisible by 3 is infinite. There are infinitely many palindromes for each possible sum of digits that is a multiple of 3."
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This tells us that even if you fix the sum of digits, you can create infinitely many unique palindromes which satisfy the divisibility rule for 3.
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For instance, take the basic sum of digits divisible by 3. Even for a small value, you can generate many different 7 digit palindromes using different permutations of the same.
- Let's assume that we have sum of 9.
- One example of a palindrome that adds up to 9 is:
- 1007001 -> sum is 9 which is divisible by 3.
- 2005002 -> sum is 9 which is divisible by 3.
- 3003003 -> sum is 9 which is divisible by 3.
- ... etc
- Notice that we can change the values of 'B' and 'C' as long as we are careful with keeping the sum at a multiple of 3. There are several different possibilities here which can make the number infinite.
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There are also infinitely many ways to construct palindromes with different digit sums which are multiples of 3.
- We could choose the sum to be 6 or 12 or 15 or any other multiple of 3 and still have infinitely many ways of constructing a palindrome based on the given pattern ABCDCBA.
Conclusion
Therefore, since we can generate an unlimited number of 7-digit palindromes where the sum of the digits is a multiple of 3, there is an infinite number of such palindromes.
