This is the first of two short blog posts today.
My younger son and I started the section on divisibility rules in our Introduction to Number Theory book. I don’t remember the context, but we have talked a little bit about divisibility rules before. He knows some of the rules, but now we are going to learn how to understand these rules through the lens of modular arithmetic.
Last night we talked about divisibility rules without even looking at the book. I just wanted to hear what he had to say:
Today we talked a little more in depth about some of the basic rules – namely divisibility by 2, 5, and 10. He seemed to be able to understand the ideas and gave a really nice explanation of why the divisibility rules for these numbers work. It was fun to hear his explanations (despite my stumbling explanation of the problem that we were working on . . . .):
I remember being fascinated by these divisibility rules as a kid, though I’m sure that I just learned the rules without really understanding why they worked. Learning the ideas behind these rules isn’t too complicated, though, and hopefully helps build up number sense and a little bit of sense about place value, too. Definitely a fun little project.
Mike's Math Page 
Does your son know if p and q are primes and a divides p*q, the a divides either p or q? With that, you can also talk about why it is necessary to check m digits.
For an extension, you could expand to do divisibility rules in other bases.
Also, for relatively simple rules 3, 9, 11 are great. A more complex one is 7. I haven’t thought about powers of those. can also play with trying to make a checksum using those rules.