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## 'Napier's Location Arithmetic' printed from http://nrich.maths.org/

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 $111
.\ldots111111$ 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 $2^3 - 1$ ($7$)