A palindrome is a word, phrase, or positive integer that reads the same forwards and backwards. For example, "kayak" is a palindrome. The integers \(292\), \(11\), and \(6\,357\,536\) are also palindromes.
Determine all the five-digit palindromic integers that are divisible by \(55\).
Note: An integer is divisible by \(11\) exactly when the alternating sum of its digits is divisible by \(11\).
To find the alternating sum, start with the first digit and alternate subtracting and adding the remaining digits from left to right. For example, \(36\,784\) is divisible by \(11\) since \(3-6+7-8+4=0\), and \(0\) is divisible by \(11\). However \(74\,253\) is not divisible by \(11\) since \(7-4+2-5+3=3\), and \(3\) is not divisible by \(11\).
Theme: Number Sense