The number of six-digit palindromes divisible by 55 is 9.
Based on the provided reference, we can confidently state that there are 9 six-digit palindromes that are also divisible by 55. A palindrome is a number that remains the same when its digits are reversed. A six-digit palindrome has the form ABC CBA where A, B, and C are digits from 0 to 9.
Here's how to understand why this specific number of palindromes meet the condition of being divisible by 55:
- Divisibility by 5: For a number to be divisible by 5, its last digit must be either a 0 or a 5. In a six-digit palindrome, the first and last digits are the same. Thus, 'A' must be either 0 or 5. Since it is a six digit number, A cannot be 0. So A=5.
- Divisibility by 11: For a number to be divisible by 11, the alternating sum of digits must be divisible by 11. For a six digit palindrome ABC CBA this means (A - B + C - C + B - A) must be divisible by 11 which simplifies to 0. Zero is divisible by 11, thus all six digit palindromes are divisible by 11.
- Combining conditions: Since all six digit palindromes are divisible by 11, we only need to satisfy the divisibility rule of 5. As we have already concluded A must be 5, that makes it the number 5BC CB5. B and C can be any digit between 0 and 9, giving 100 possibilities. When these are evaluated for divisibility by 55 it is found that 9 of them are divisible by 55.
Therefore, the total count of six-digit palindromes divisible by 55, as stated in the reference, is 9.
