Of Number Plates and Book Numbers

by Tawanda Gwena

First the preliminaries. A number m modulo n is obtained by subtracting as many n's out of m as possible, where m is usually a whole number. A shorter method would be to divide m by n and take the remainder.

Now the problem is how we can use it. In this article we will give two applications of this idea. One national and one international.

In Zimbabwe every car has (or should have) a number plate. The government plates are not interesting here, so I will stick to the ordinary numbers of the the form 3 digits dash three digits letter. The number itself is not very interesting, as these are just handed out by the people who hand them out. The number is incremented by one for each person. At the moment the top number is about 660-000, so we can estimate there are about 600 000 cars in the country. The interesting bit here is the letter. It is not just handed out, but is calculated from the digits. Its main purpose there is to make sure the digits are correct. It is called a check digit, even though in this case it is a letter. They work it out as follows. Leave out the hyphen and take the digits as one long number N. Divide it by 23 and take the remainder. With the following table you can then get the letter. In symbols the number of the letter is N mod 23.

They divide by 23 because the alphabet they use has only 23 characters. The letters i, o and u are missing because they might be confused with the number 1, the number 0 and the letter v respectively.Now that you know how to forge number plates, find out which of the following are genuine: 649-032T, 539-524R, 417-574B. (Answers at the bottom of the page)

(I suspect that the letters for national ID numbers are done in the same way, if you ignore the last two numbers in them. Check this yourself.)

The second application is in reference to books. Every book published in the world is given what is called an International Standard Book Number or ISBN for short. The number itself is interesting but at the moment let us concentrate on the last digit. This is a real check digit (it is almost always a digit) as it is derived from the others.

We begin with an example. Take the book ``LaTeX, A Document Preparation system'', by L. Lamport with the number 0-14-017997-6. Your working is as follows. Ignore the last digit -- that's the one we want to work out from the others. Now, multiply each digit by its position in the number, add the results up, and find the remainder on division by 11. In this case we have

(0 x 1) + (1 x 2) + (4 x 3) + (0 x 4) + (1 x 5) + (7 x 6)+ (9 x 7) + (9 x 8) + (7 x 9) = 259

This, on division by 11 gives a remainder of 6, which is our last digit! That is how the method works. In the case when the remainder is 10 the letter X is written.

This complicated system has a very good reason for existence. When such numbers are transmitted errors can occur. If an error has occurred it will usually create an inconsistency in the last `checking' digit, which can be spotted by computer and corrected.

For more information on ISBNs, check out its entry in the Weston's Encyclopedia.

As for the problem given earlier, only the first number plate is genuine.