h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

I cannot recall having any degree in maths. The latest brainteaser involving binary code and codewords is beyond my knowledge, but I do know about binary numbers.
Getting back to my brainteaser and the easy method of solving it.

It is a known fact that any number which is a multiple of 9, added to, subtracted from, multiplied by, or divided by another number which is a multiple of 9, will result in an answer which is a multiple of 9.
It follows that any number which is 2 more than a multiple of 9 taken away from a number which is 6 more than a multiple of 9, must result in a number which is 4 more than a multiple of 9.
If added the answer would be (6+2)8 more than a multiple of 9.
If multiplied the answer would be (6 times 2) 12 reduced to 3 more than a multiple of 9.
The same principle applies to division.

By adding all the digits in any number, and reducing to a single digit, tells you whether it is divisible by 9, and, if not, what the remainder will be.

6258259(37)(7+3)(10)1

Thank you very much for the explanation.
It is beautifully simple, with a twist of logical genius.

One of those things where one will never get to the solution unless one happens on the correct angle at which to look at it.

Maybe you are related to Lewis Carroll?