There are 30,000 seven-digit palindrome numbers that are also divisible by 3.
Understanding Palindromes and Divisibility by 3
A palindrome number reads the same forwards and backward. A seven-digit palindrome has the form ABCD CBA, where A, B, C, and D are digits. Since it is a seven-digit number, A cannot be 0. Therefore, A can be any digit from 1 to 9, while B, C, and D can be any digit from 0 to 9.
A number is divisible by 3 if the sum of its digits is divisible by 3. In our case, the sum of the digits of the palindrome ABCDCBA is A + B + C + D + C + B + A = 2A + 2B + 2C + D = 2(A + B + C) + D.
Calculating the Number of Palindromes Divisible by 3
For the palindrome to be divisible by 3, 2(A + B + C) + D must be divisible by 3. This is equivalent to saying that D must have the same remainder as -(2(A+B+C)) when divided by 3. Since we're working with remainders modulo 3, this is also equivalent to D having the same remainder as (A+B+C) when divided by 3.
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Choosing A, B, and C: A can be any of the 9 digits from 1 to 9. B and C can each be any of the 10 digits from 0 to 9. So, there are 9 10 10 = 900 possibilities for the digits A, B, and C.
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Determining D: Once A, B, and C are chosen, the sum A + B + C will have a specific remainder when divided by 3 (0, 1, or 2). For each remainder, there are either 3 or 4 possible choices for D (0 to 9) to make 2(A+B+C) + D divisible by 3. More specifically, if A+B+C has remainder 0 mod 3, then D can be 0, 3, 6, or 9. If A+B+C has remainder 1 mod 3, then D can be 2, 5, or 8. If A+B+C has remainder 2 mod 3, then D can be 1, 4, or 7. Therefore, there are on average 10/3 possible digits for D, which we will approximate as the exact answer of 10/3.
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Calculating the Total Number: The total number of 7-digit palindromes divisible by 3 is therefore (9 10 10) (10/3) = 900 (10/3) = 3000.
Examples
Let's consider some examples:
- 1000001: 1 + 0 + 0 + 0 + 0 + 0 + 1 = 2. Not divisible by 3.
- 1020201: 1 + 0 + 2 + 0 + 2 + 0 + 1 = 6. Divisible by 3.
- 1110111: 1 + 1 + 1 + 0 + 1 + 1 + 1 = 6. Divisible by 3.
Conclusion
Therefore, there are 30,000 seven-digit palindrome numbers that are also divisible by 3.
