#math trivia #171 solution

The sum of the digits of a number divisible by 9 is itself divisible by 9; this follows from the fact that powers of 10 are always one more than a multiple of 9.

For the solution, let’s look at one-digit, two-digit, and three-digit numbers separately.

  • One digit:  All one-digit numbers are palindromes; the only one divisible by 9 is 9.
  • Two digits:  The only two-digit numbers that are palindromes are the ones divisible by 11.  The only one also divisible by 9 is 99.
  • Three digits:  Any three-digit number of the form aba is a palindrome.  The only ones divisible by 9 are those for which a+b+a = 2a+b is divisible by 9.  With the limitation to day-numbers, we know that a must be 1, 2 or 3.  This limits the answers to 171, 252, and 333.

Answer:  9, 99, 171, 252, 333.

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