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John Grant McLoughlin offered this problem in Crux Mathematicorum in April 2008:

Every four-digit numerical palindrome (e.g., 2772) is a multiple of 11. Why is this?

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Any such number can be written in the form abba, where a and b are integers, with a ranging from 1 to 9 and b from 0 to 9. But abba = 1000a + 100b + 10b + a = 1001a + 110b = 11(91a + 10b). So 11 is always a factor.