Talking about some introductory number theory ideas with the boys

My younger son is working his way through Martin Weissman’s An Illustrated Theory of Numbers right now:

He’s in the chapter on greatest common divisor and least common multiple now, and I thought talking through some of the ideas he’s seeing would make for a good project this morning. It gave him a chance to talk about what he’s learning and it gave my older son a chance to review some ideas he’s seen before.

We started by talking about the Euclidean Algorithm:

Next we discussed the interesting identity that the product of the LCM and GCD of two positive integers is equal to the product of those integers:

Now we moved on to discussing how the ideas we talked through in the prior videos could help us solve Diophantine equations. Here my younger son introduces the main ideas:

To finish, I had my older son explain why the general solution my younger son introduced in the first video was

I can’t say enough good things about Martin Weissman’s book – it has really gotten my son interested in number theory. Can’t wait to explore more of the ideas in the book with him!