The final part of this problem is very challenging, but students can gain much that is valuable just from Part One. Validating the process as a reliable method of multiplication involves some clear reasoning and communication skills.

The second and third parts of this problem are included as a challenge for abler students and intended to draw them into a deeper understanding of the structure.

One possible approach for classroom use could be to ask what factors could produce particular products. Students could, for example, create products and challenge each other to find possible factors.

There are also patterns that suggest structure. For example 111 . . .111111 in binary sometimes has a factor and sometimes does not, depending on the length of the line of ones. 1111 is the product of 11 and 101 but 11111 has no factors.

Teachers may wish to offer students, as a conjecture to explore, the suggestion that

2^{ab} - 1

has

2^{a} - 1

as a factor.

For example : $2^{6} - 1$ is 63 and its factors include $2^{2} - 1$ (3) and $2^{3} - 1$ (7)