Why Does This Weird Number Trick Work?

A number minus that number but with its digits rearranged gives a result that’s divisible by 9

So I saw this tweet from Presh Talwalkar, which states the following:

So we take a random number (an integer) — say 74356— and rearrange it to e.g. 46375. We subtract the latter from the former and get 74356 – 46375 = 27981. And 27981 is indeed divisible by 9: 27981 / 9 = 3109.

This works for any integer, which Presh Talwalker shows. It’s an interesting read, and I’d like to offer my own approach as well.

Let’s look at the previous example again: 74356 – 46375. First — and this may look weird — we subtract 74356 from itself.

Of course, we get 0. That is, of course, not our answer to 74356 – 46375. But how much, exactly, does our answer (0) differ from the right one?

Let’s look at the first step:

In 46375, the 6 is “worth” 6000. So in this step, we should have subtracted 6000, but only subtracted 6. So we should subtract an extra 6000 – 6.