For any multiple of 37, reverse and interleave with 0s, you get a multiple of 37

Reminds me of the "magic" trick where they ask you to write down a three-digit number, and then subtract it from those same three digits reversed. (And then take that number and add it to itself with its three digits reversed.) It's always 1089. (So then the magician tells someone to look on page 1089 of a book, where they've written "Your answer will be 1089!" or some such...)

I tried explaining this to a friend once...

Reversing the digits means that the last digit becomes 100-times-the-last-digit -- so when you subtract, you end up with 99 times the last digit. And it's the same for the digits on the other end, except it ends up being negative-99-times that digit. The middle digit always stays the same, so subtracting it from itself always adds 0 to the total, and you always end up with something that could be expressed as 99x - 99y -- which is 99 * (x-y). All that's really important is you end up with a multiple of 99, since all multiples of 99 have a pattern. The middle digit is always a nine, while since you're always one short of a hundred, the first digit goes up by a hundred each time you add 99 -- while the last digit goes down by one. So your first and last digit will just always add up to nine. Meaning that adding any of these numbers to its reversed-digits counterpart always gives you 900 plus 90 + 90, plus 9 -- or 1089.

You can also totally screw this up for the magician if you use the same number for the first and last digit. (Because then the result of the first subtraction is.... zero.) It's still a multiple of 99 -- but it's zero times 99.