Open ended: Creating Arrays

In this project, students attempt to create all of the possible array representations for each of the numbers smaller than a given number(which may depend on the amount of time they have to devote to this project).

For example 6 = 2 x 3 and 6 = 1 x 6, so two possible arrays for 6 are:

***
***

or

******

The objective after they have these arrays is to classify the numbers into groups based on whatever patterns they see in the structure of the arrays.

What observations do you think students might have when doing this project?

Open-ended: Perfect Numbers

A number is called perfect if the sum of the proper divisors is equal to the number. 8 is not a perfect number because the proper divisors of 8 are 1, 2, and 4, and 1 + 2 + 4 = 7, which is not 8.

Can you find a perfect number? Can you find all of the perfect numbers less than 1000? Could you write a program to find all of the perfect numbers smaller than a million? Is there a largest possible perfect number?