A divisor is a number that divides into another number without leaving anything left over. Let's compare ten and twelve:

• two is a divisor of ten, so is five. That's all the divisors ten has which as we'll see later is somewhat of a bummer for us humans.
  • 10 / 2 = 5
  • 10 / 5 = 2
  Three of course is not a divisor of ten since ten divided by three is three with one left over.
• two is also a divisor of twelve (in fact, a number is even if it has two as a divisor). Three, four, and the number six are also divisors of twelve which, unlike ten's paltry collection, is a good thing as we'll see shortly.
  • 12 / 2 = 6
  • 12 / 3 = 4
  • 12 / 4 = 3
  • 12 / 6 = 2

Let's look at what division is, not from a technical mathematical point of view but the visceral reality of division. Division is splitting something into chunks. In the scope of what we're talking about we'll restrict ourselves to equal-sized chunks, and ways to represent those chunks.

Given a task of splitting something into equal-sized chunks ideally it'd be good (since one of our premises is that humans like it easy) if it were simple & convenient to express what we've been left with after our splitting efforts. As we just saw with splitting ten into two equal-sized chunks it was easy to then tell someone what we've done and what we're left with: "Hey Joe, I split this pack of ten into two, five in each chunk". Same with splitting it into five chunks, two in each.

Now, try splitting ten into four equal sized chunks, you're left with bits that are sized two-and-a-half.

Uh oh!

In order to express that we've had to use fractions. In a numbering system based on ten, that's 2.5₁₀. Not too bad though. It gets worse quickly however: suppose we wanted to express what we'd done when we split that ten into three: each of the chunks is three-and-a-third. How do we write that without using fractions?

Double uh oh!

There's no way to express that in base ten! It's 3.33333...₁₀ and recurring forever on. Actually there is some special notation for expressing recurring decimals like that but it is in essence really just a long way of writing out fractions: at the end of the day you simply can't express the result of something as apparently simple as splitting ten into three chunks.

So we've looked at how to express the results of splitting the number ten into two, three, four and five chunks. Only two and five, the divisors of ten, were expressible as integers (by definition, if you think about it), four was "OK" and three was inexpressible. Twelve by contrast has four divisors, two, three, four and six. In other words, you can split twelve into equal-sized chunks of those numbers and be able to express the result without fractions or floating points. What about five? Twelve divided by five is two and two-fifths but this is hard to express in base twelve. It's one of the few numbers that doesn't mesh well in base twelve.

We've looked at what a divisor is, what division is at its basic level and the relative difficulty of telling someone the results of your dividing activities. Twelve certainly seems to have the edge but by exactly how much? Our grand finale is a full comparison of base ten and base twelve doing division...