There are 30 four-digit palindromic numbers divisible by 3.
This answer is based on the provided reference stating: "There are 90 four-digit palindromes from 1001;1111;1221;1331;... to 9669;9779;9889;9999. Because (12;15;18;21;...;93;96;99) are divisible by 3 and this set has (99−12)÷3+1=30 numbers, there will be 30 palindromes satisfy the task." This calculation focuses on four-digit palindromes. To determine the number of palindromes divisible by 3 for other lengths (e.g., 5-digit, 6-digit, etc.), a similar approach, considering the divisibility rule for 3 (sum of digits divisible by 3), would need to be applied.
Understanding Palindromes and Divisibility by 3:
- A palindrome is a number that reads the same forwards and backward (e.g., 121, 1331, 5445).
- A number is divisible by 3 if the sum of its digits is divisible by 3.
Example (Four-Digit Palindromes):
Let's consider a four-digit palindrome represented as abba, where a and b are digits. The sum of its digits is 2a + 2b. For this sum to be divisible by 3, 2(a + b) must be divisible by 3. Since 2 and 3 are coprime (have no common factors other than 1), this means (a + b) must be divisible by 3.
The reference directly provides the calculation and result for four-digit palindromes. Extending this to other lengths requires a different calculation based on the number of possible palindromes and the divisibility rule for 3.
