# A nice divisibility rule problem from James Tanton

Saw this fun puzzle from James Tanton today:

By luck, my younger son and I just finished the section about divisibility rules in our Introduction to Number Theory book, so this was a well-timed puzzle.

When I got home tonight we looked at the puzzle together – starting with a quick review of the divisibility rule for 11. It was also nice to see that my son was able to count the number of 5 digit palindromes since I thought that we might have to spent a little extra time on that part:

Next we took a first crack at counting the number of 5 digit palindromes that are divisible by 11. We got really unlucky when we were filming this video and the camera’s memory got filled up about 2 minutes in the first time through. Sorry if this one seems like it starts too quickly – if it seems that way it was because we were starting over 😦

The unlucky start start aside, though, this video shows why I think there’s a great project in here for kids – so much secret little algebra and arithmetic practice hiding in this problem. Lots of interesting patterns, too!

The next part was counting the number of 5 digit palindromes that have the alternating sum of their digits equal to 11. There’s a fun little pattern on this part, but seeing it involves going through several different examples. Lots of great algebra and arithmetic practice, though, and I was really happy to see that my son stayed engaged all the way through – even taking a couple of guesses at the patterns.

Now we were in the home stretch! In this part we count the number of palindromes with alternating sum equal to zero. The surprise in this section is that the pattern is similar to the one we saw in the last video. Why are the patterns for summing to 11 and summing to 0 related? Hmmm . . . is there more going on here than James Tanton is letting on?

The last step was checking when the alternating sum was equal to -11. As my younger son guessed, we find a similar pattern to the case when the alternating sum was equal to 22. Fun!

So, we find that a total of 6 + 35 + 35 + 6 = 82 5-digit palindromes that are divisible by 11. As I said above, the timing of James Tanton’s tweet couldn’t have been better since we just talked about the divisibility rule for 11. With that rule in our back pocket, this project became a really great example of how to put that rule to work and also get lots of fun arithmetic and algebra practice. Another great problem from James Tanton!

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