The sum of the divisors of an integer

Today we can to the end of our short project on divisors. The topic for today was finding a formula for the sum of the divisors of an integer.

I wasn’t entirely sure how to go about explaining this idea to the boys – for one thing a thorough explanation sort of requires some geometric series formulas – so I just let them take the lead.

We started by reviewing the formula for the product of the divisors and then began to talk about the sum. I posed the question about the sum as a challenging math problem with the specific idea of thinking about how to break this problem down in to smaller pieces that we could understand.

In the last video we found formulas for the sum of divisors of a prime and the square of a prime. We kept going with that idea at the beginning of this video and then the boys started noticing a pattern:

At the end of the last video we started forming an idea about the formula for the sum of the divisors of a number. We spent another 5 minutes exploring that formula before moving to Mathematica:

Finally, we went to Mathematica to play around with other numbers a little more quickly.

So a fun week with a little number theory and a little computer math. The project this week were sort of inspired by a “things you should know list” from my older son’s school math team. Although I was a bit surprised by some of the topics on that list, talking through a few of those topics this week was really fun.

A nice problem for kids from James Cleveland

Saw this problem posted on twitter by James Cleveland:

I thought it would make a nice problem for the boys to talk through. It was actually much better than I expected from the point of view of hearing kids talk about math.

So, without much comment, here are their thoughts. My 6th grader first and then my 4th grader. Probably the most difficult piece for both kids was figuring out that the two solutions they found were, in fact, the only two solutions.