MAT 105 Spring 2008

The check digit systems we will study are used
for:
 US Postal Service money orders
 Airline tickets
 UPC (Universal Product Code)
 US Bank routing numbers
 Credit card numbers
 ISBN (International Standard Book Number)
The ID number
is listed here
The ID number is also listed here in
machine-readable numbers (magnetic ink)

The ID number on a USPS money order is an
11-digit number, and the 11th digit is the
check digit

The 11th digit is the remainder when the sum
of the first 10 digits is divided by 9

In our sample money order, the ID number is
02543750594

If we add up the first 10 digits, we get 40, and
the remainder when 40 is divided by 9 is 4, so
the check digit is correct

Another example: 63024383845

Since many of the check digit systems involve
finding remainders, it is useful to know how
to find them on your calculator

There are many different methods, but this
one is simple

For example, suppose you need to know the
remainder when 59 is divided by 7

To find the remainder when 59 is
divided by 7, just type 59 divided
by 7 in your calculator

Take the digits appearing after the
decimal and multiply them by 7
(the number you divided by)

The result will be the remainder

In this example, the remainder is 3

Suppose we receive a suspicious money order
with ID number 63054383845

If we add up the first 10 digits and divide by 9,
we get remainder 8, which does not match
the check digit

So we know this ID number is invalid

Look at what happened:
 Valid ID number
 Invalid ID number
63024383845
63054383845

This is a substitution error: an incorrect digit
was substituted for the correct one

This error was detected because we were
able to tell that the new number is invalid

Let’s look at another example
 Correct ID number
 Incorrect number
63024383845
63924383845

Notice that the incorrect number is actually still a
valid ID number, so this error goes undetected by
the check digit system

In fact, this system can never detect a
substitution of a 0 for a 9 (or vice versa)

Since we just add up the first 10 digits, this
system is also unable to detect transposition
errors
 Correct ID number
 Incorrect number

63024383845
63023483845
Once again, the incorrect number is still valid
This is the ticket ID number. The last digit
(colored in yellow) is the check digit.

The check digit is the remainder when the ID
number (without the check digit) is divided by 7

It is difficult for us to find these remainders on
our calculators when the ID numbers are very
large, like they are on airline tickets

For our examples, we will use ID numbers that
are shorter than normal, just to illustrate how
the process works

Is the airline ticket ID number
5208162 valid?

Remember, 520816 is the ID
number, and 2 is the check digit

On our calculators, we divide
520816 by 7 and get remainder 2

This method detects all single substitution errors
except 0  7, 1  8, and 2  9

In addition, this system can detect transpositions as
long as the two digits are not 0 & 7, 1 & 8, or 2 & 9

Examples




5208162  5204162 detected
5208162  5201162 not detected
5208162  5280162 detected
5208162  5201862 not detected

Remember how error detection works:
 If we change the ID number and now the check
digit is wrong, the error is detected
 If we change the ID number and the check digit is
still correct, the error is not detected

Make sure you understand the difference
between “incorrect” and “invalid”