Dave Radcliffe’s amazing Fibonacci / GCD post

This morning I received a neat heads up on a really cool post from Dave Radcliffe on twitter:

I’ve been covering the concept of greatest common divisor with my younger son this week, so Dave’s post had some great timing.  It is such a neat formula that I thought I’d use it for a little talk with my kids tonight instead of doing our usual weekend Family Math.

We started by talking through Dave’s formula and writing down the first 15 Fibonacci numbers.  I then worked through a simple example to make sure that the kids understood the language of the formula:

next I had my younger son see if he could work through the formula using the 8th and 12th Fibonacci numbers.  He’s just learned the Euclidean algorithm, so I was assuming that he’d use that approach to find the greatest common divisor.  This formula turns out to be a great way to practice the Euclidean algorithm.

I wanted my older son to work through an example, too, and picked one with slightly larger numbers – the 10th and 15th Fibonacci numbers.  I have covered a bit more number theory with him and suspected that he would remember that you could find the greatest common divisor of two numbers by factoring.  I was glad that he did since it was a nice example for my younger son to see:

Finally, to do a few more complicated examples we went to the computer.  Since Mathematica has built in functions to calculate the GCD of any two numbers as well as to calculate any Fibonacci number, we could easily work with much larger numbers.  We did a couple of examples, including one where we forced a large greatest common divisor.

I’ve given no thought at all to how to prove Dave’s formula, but I do have a little bit of driving to do this weekend so at least I’ll have something to think about!  The boys seem to enjoy seeing fun little math facts like these, so despite not having a lot of context around this amazing formula I’m happy with how this project went.  Definitely not your standard greatest common divisor exercise!

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