Twitter Math Puzzle and Solution

Yesterday I posted a very simple math puzzle to Twitter that I found in Jonathan Baron’s book, Thinking and Deciding. The puzzle is the following:

Show that every number of the form ABC,ABC is divisible by 13.

The puzzle comes up in Baron’s book as an example of an “insight problem” in which one goes from not knowing the answer at all to knowing the complete answering in a sudden moment of insight.

Several people replied to my tweet with solutions: I especially like Will Townes’s solution. In particular, if you’re familiar with modular arithmetic, I like the logic of Will’s answer because it gives a simple generalization. First, represent ABC,ABC as ABC * 1000 + ABC * 1 rather than as ABC * 1001. Then notice that

  1. 1 = 1 mod 13
  2. 1000 = -1 mod 13

Thus ABC,ABC = ABC * -1 + ABC * 1 = 0 mod 13. This logic can be easily extended to show that (ABC,ABC,)*ABC,ABC = 0 mod 13 no matter how many times you repeat the ABC,ABC pattern.

8 responses to “Twitter Math Puzzle and Solution”

  1. Z

    Isn’t it just enough to notice that ABCABC = 1001 * ABC and that 1001 = 0 mod 13?

  2. alex

    or without modular arithmetic: abcabc = abc * 7700 * 13 is obviously divisible by 13.

  3. Mikko


    ABCABCABCABC is also simple enough (and similarly for an even amount of ABC): since ABC,ABC,ABC,ABC = 10^6*(ABCABC) + (ABCABC) = 0 mod 13 since ABCABC is.

  4. Mikko

    Or rather, ABCABC,ABCABC is divisible by 13 since ABCABC is (and similarly for an even amount of ABC). I just think using modulo arithmetic is totally overkill for a simple problem like this one.

  5. bob

    mikko, how can you say that modular arithmetic is totally overkill? you seem to imply that some maths are more inherent than others.

    the author only seemed to say that he liked the modular solution, not that it was (in some arbitrary viewpoint) the most basic solution; if he likes a piece of math because it shows the answer to the problem in an interesting way, that’s great.

  6. Will Townes

    Hi John,
    I ran into an old colleague today and he told me he had seen my answer here on your blog. I didn’t even realize you had posted it, so thanks for indirectly reconnecting me with him! I’m flattered that you enjoyed the solution enough to post it. It’s always a pleasure to learn from your ideas.
    Best wishes,
    Will Townes