Chapter 16: Check Digit Systems, Part 1
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Transcript Chapter 16: Check Digit Systems, Part 1
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”