I saw an old note in our Number Theory book about a problem I used for a movie with my older son. Seeing that note made me want to try the same problem today with my younger son and then go and look at the difference two years had made.

The problem is: Let N be a positive integer with exactly 11 divisors. How many divisors does have?

From October 17, 2012:

From October 14, 2014:

Similarities:

(1) Both kids recognize that only perfect squares can have an odd number of divisors.
(2) Both kids recognize the formula for calculating the number of divisors. I found it interesting that my younger son seemed to understand the idea behind this formula at the beginning of the video, but the large power at the end of his video gave him a little trouble.

(3) Both kids recognize that 11 is prime and hence is only divisible by 11 and1, and seem to understand how to use that idea to solve the problem.

(4) I do more work in the first video compared to the second, so I’ve learned to slow down a little which is nice.
(5) Both kids recognize some basic powers about powers, though my older son seems more comfortable with those concepts.

(6) and Ha – we had a fire in the fireplace on October 17th 2012 and on October 12th 2012. Some things don’t change 🙂

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