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 mod 13
- 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.