Go grab some dot paper or grid paper—or just make some dots in a square grid on a blank piece of paper. Let’s start with a 4 × 4 grid of dots, like so.
Now, start at the top left corner, draw a vertical line down to the bottom of the grid, and count each dot that your pen enters—which just means that you won’t count the first dot, since your pen leaves that dot but does not enter it. Then, draw a horizontal line to the right, starting over with your counting. Again, count each dot that your pen enters. Count just 2 dots as you draw to the right.
Finally, draw a straight line (a hypotenuse) back to your starting point. Here again, count the number of dots you enter.
One example is not, of course, enough to convince you that the number of dots your pen enters when drawing the hypotenuse is the greatest common factor (GCF) of the number of counted vertical dots and the number of counted horizontal dots. So, here are a few more examples with just a 4 × 4 grid.
No doubt there are tons of people out there for whom this display is completely unsurprising. But it surprised me. The GCF of two numbers is an object that seems as though it should be rather hidden—a value that may appear when we crack two numbers open and do some calculations with them, not something that just pops up when we draw lines on dot paper. We use prime factorization to suss out GCF, after all, and that is by no means an intuitive process.
Connections
There are some very nice mathematical connections here. The first is to the coordinate plane, or perhaps more simply to orthogonal axes, which we use to compare values all the time—but only in certain contexts. Widen or eliminate the context constraint, and it seems obvious that comparing two numbers orthogonally could yield insights about GCF.
And slope is, ultimately, the “reason” why this all works. The slope of a line in lowest terms is just the rise over the run with both the numerator and denominator divided by the GCF: \[\mathtt{\frac{\text{rise}}{\text{run}}\div\frac{\text{GCF}}{\text{GCF}}=\text{slope in lowest terms}}\]
Once slope is there, all kinds of connections take hold: divisibility, fractions, lowest terms, etc. Linear algebra, too, contains a connection, which itself is connected to something called Bézout’s Identity. There is also a weird connection to calculus—maybe—that I haven’t quite teased out. To see what I mean, let’s also draw the LCM out of these images.
From the lowest entered point on the hypotenuse, draw a horizontal line extending to the width of the triangle. Then draw a vertical line to the bottom right corner of the triangle. Now go left: draw a horizontal line all the way to the left edge of the triangle. Then a vertical line extending to the height of the lowest entered point on the hypotenuse. Finally, move right and draw a horizontal line back to where you started. You should draw a rectangle as shown in each of these examples. The area of each rectangle is the LCM of the two numbers.
The maybe-calculus connection I speak of is the visible curve vs. area-under-the-curve vibe we’ve got going on there. I’m still noodling on that one.