Not surprisingly, today’s Family Math began with a neat question that James Tanton posted on Twitter:

I chose to not go through the entire problem but thought it would be fun to answer a slightly easier question – what is the smallest positive integer with exactly 100 divisors?

We started by talking through the problem and looking for some ideas on where to start. Turned out that each kid had an interesting idea, and I thought it would be fun to pursue both of them:

We started off down the path suggested by my younger son – start by looking to see if there are any patterns in the number of divisors on the integers. The difficulty is that although there is a pattern, in fact a really cook pattern, it is a little tough to see. I let them noodle on it for a while and then asked them to talk a little bit about what it means to be a divisor of a number. That led to my older son noticing a similarity in the prime factors.

Next we moved to looking at the factors of a larger number -> 24. After understanding the divisors of 24, we built off of the counting project we’ve been doing this summer to see how to find an easier way count the number of factors from the primes. Making this connection was the reason I wanted to do this little project.

My younger son’s idea led us to the rule for counting the number of divisors of of a given integer. Next we move on to my older son’s idea of looking for a pattern when we list out the smallest integers with a given number of factors. To aide in this part of the project we used Mathematica and had a lot of fun looking at lots of patterns in the numbers (also sorry about the camera issue in this video):

Having checked what the smallest integer was that had N factors for N ranging from 1 to 10, we went back to the whiteboard to look more carefully at those numbers. Here my lack of planning ahead came back to bite me a little since the prime factors of all of these numbers had only 2’s and 3’s. To correct for that I added in the smallest number with exactly 12 factors – that integer is 60.

Talking through the pattern we see in this list lets us take a couple of guesses at the smallest integer with exactly 100 factors. Looking at our guesses, the smallest one is

Having gone through the difficult part of the problem, we head back to Mathematica to see if our guess is correct. Mathematica gives us a short list of numbers with exactly 100 factors, and our number is indeed the smallest!

Finally, we wrap up by taking a quick look at the second smallest number with 100 factors. When we factor it into primes we see that it looks pretty similar to the number we found – just one prime factor is different.

When you see the original problem for the first time it seems almost impossible to solve. A little bit playing around with patterns leads to the amazing discovery that the problem isn’t as intractable as it seems, though. You also get a look at several really cool patterns relating integers to their prime factors. Definitely a fun project and a fun way to show how computers can be helpful in solving problems, too.