Counting divisors and and some sneaky arithmetic practice

My younger son chose a neat problem to discuss from Art of Problem Solving’s Introduction to Number Theory book today. It is problem 3 from section 5.2 in the book.

I like the problem for two reasons. First, this entire book gives young kids a great chance to review arithmetic while learning some ideas in math that aren’t typically part of an elementary school curriculum. Second, many of those ideas are really cool!

Here’s the problem plus his arithmetic-based solution:

 

It turns out that this problem can be solved without listing all of the divisors, which is sort of amazing. The next section in the book goes through the details of how to do that, but I thought it would be fun to talk through the idea using this problem. It is a fun way to show off the power of math – it is really surprising (especially to a young kid!) that you can solve the problem without actually listing out all of the divisiors!