The math team for my son’s new school has a little brochure telling new students some of the ideas that come up in math contests. I don’t know if this list is something that the kids are supposed to know ahead of time, or a list of things that they’ll be talking about during the year, but I thought it would be fun to spend the week touching on a few of the ideas. Today’s topic fit right in with our summer counting project: counting divisors of a number.

We’ve done a few divisor counting problems in the past, so I hoped this would be an easy review topic for the kids. Here are three of our old projects:

The Same problem two years apart

A neat number theory problem for kids from Tracy Johnston Zager

A neat number theory problem for kids from David Radcliffe

We talked about how to count divisors for maybe 15 minutes and then turned our attention to problem 5.29 from our well-worn copy of Art of Problem Solving’s *Introduction to Counting and Probability* book.

The first part of the problem is here:

Now that we’ve found the minimum number of primes, what about the maximum:

Next up – what is the smallest positive integer with exactly 20 divisors?y

Last up – the challenge problem: Is there a positive integer smaller than 240 with more than 20 divisors. I was really happy to hear the ideas that the boys had while trying to solve this problem:

After the long discussion on the challenge problem, we went to play around with the question with Mathematica. The kids were incredibly engaged when they saw the various ways we could study the problem (and similar problems) with the computer:

So, a fun project for kids inspired by the math club at my son’s new school and also by Art of Problem Solving’s awesome *Introduction to Number Theory* book.