Notes On Divisibility

Imagine the numbers 1, 2, 3, … laid out in a regular linear sequence, like mile markers on a long road.

mile marker 258, marking 258 miles from the beginning

mile marker 258, marking 258 miles from the beginning

The markers don’t need to be a mile apart, you can put them closer if you like, but you will want to put them at a regular distance.  In this arrangement, certain basic patterns can be highlighted by touching and skipping markers.  For starters, I will go on a 2-tour: I will touch every second mile marker, but skip the ones in between, so I touch all the markers 2, 4, 6….  I could scribble a small red “2” on each of those, as a record of my trip.  The red “2” will tell me later, at a glance, if a particular marker was touched by me on on my 2-tour.  After the 2-tour, I can cover the same territory later, and do a 3-tour, where I start once again at the beginning but touch ever third mile marker, skipping two markers in between.  These markers that are three apart, 3, 6, 9…  are marked with a small red “3” as a souvenir of my 3-tour.  And then I can make a 4-tour, and so on.

After enough time, each mile marker will have red marks from various -tours, and if you were to look at a particular mile marker, like this 258-mile marker, you would find red marks there from precisely those tours that touched it.  On this marker you will find a red mark for the 258-tour, but of course no marks for the 259-tour, the 260-tour and up.  None of the later tours will touch the 258-mile marker, but some of the earlier tours have.  (For completeness, I could also do a 1-tour, simply touching all markers, and I mark these with a small green “1”.)

What we’ve done to the mile markers is relatively straightforward, and yet the patterns established by our red marks are surprisingly intricate and involved.  I intend to do a bunch of looking at these patterns and at what they reveal later, but for now the thing I want to emphasize is by what simple means these patterns have been laid down.  We find ourselves swimming in the sea of divisibility after just a few steps on the beach of skip counting.

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