( c )
C a r t e r ,
E p s t e i n
a n d
B o l l i n g e r
2 0 1 6
C h a p t e r
1 6 :
I d e n t i f i c a t i o n
N u m b e r s
P a g e |1
CHAPTER 16: Identification Numbers
Identification numbers should be ______________. So each number
represents a different person or thing.
List some examples of identification numbers:
Identification number often contain one or more ___________________
to help catch data entry and data transmission errors.
Modular Arithmetic (from section 17.4)
Example
If a class starts at 11:00 a.m. and
lasts for 3 hours, what time does it
end?
This is a form of modular arithmetic. Modular arithmetic is helpful when
forming check digits.
We say that a modulo m is ________________ to b (written as
𝑎 mod 𝑚 ≡ 𝑏) if (𝑎 − 𝑏) ÷ 𝑚 has a remainder of 0. This means 𝑎 − 𝑏 is
a multiple of 𝑚. One way to find a value for 𝑏 is to find the
________________ when 𝑎 is divided by 𝑚.
Example
For our clock example, 11 + 3 = 14. Because 14 ÷ 12 has a remainder of
________, 14 mod 12 ≡ _________. This also means that (14 − 2) ÷ 12
has a remainder of ___.
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C a r t e r ,
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2 0 1 6
C h a p t e r
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I d e n t i f i c a t i o n
N u m b e r s
Example
Determine if the following congruences are true or false.
a)
27 mod 9 ≡ 0
b)
29 mod 10 ≡ 9
c)
17 mod 5 ≡ 3
Example
Find the following values.
a)
17 mod 5 𝑖𝑠 _______
b)
47 mod 11 𝑖𝑠 _______
c)
53 mod 7 𝑖𝑠 _______
d)
8 mod 13 𝑖𝑠 _______
P a g e |2
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C a r t e r ,
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2 0 1 6
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N u m b e r s
P a g e |3
16.1 Check Digits
Common Data Entry Errors
________________ - Occurs when an incorrect digit is entered.
This is also called a substitution.
Ex:
________________ - Occurs when adjacent digits are entered in
reverse order.
Ex:
________________ - Occurs when digits that are separated by
another digit are entered in reverse order.
Ex:
SUM of the digits mod 9
Example
The first 10 digits of a USPS Money Order identify
the money order. The last digit is a check digit,
which should equal the sum of the first 10 digits
mod 9.
Verify that the check digit in this image is correct.
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2 0 1 6
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N u m b e r s
P a g e |4
Check digits are designed to catch errors, so it is important to know which
errors will NOT be caught.
 For an error not to be caught, the correct number and the number with
an error must produce the ___________ check digit.
 If 𝑥 𝑚𝑜𝑑 𝑛 ≡ 𝑦 mod 𝑛, then 𝑥 − 𝑦 is a _____________ of 𝑛.
 We will label our correct number with 𝑎1 𝑎2 𝑎3 … where each 𝑎𝑖
represents a ___________ digit.
 For transposition errors, we will rearrange the 𝑎𝑖 according to the
type of transposition.
 For single digit errors, we will use 𝑒𝑖 to represent the error for the
digit that should have been 𝑎𝑖 .
( c )
C a r t e r ,
E p s t e i n
a n d
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2 0 1 6
C h a p t e r
1 6 :
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N u m b e r s
P a g e |5
Example
Will the “sum of digits mod 9” check digit scheme that is used for money
orders detect all single-digit errors?
Let’s look at an error in the first digit, 𝑎1 .
Correct Code: 𝑎1 𝑎2 𝑎3 𝑎4 𝑎5 𝑎6 𝑎7 𝑎8 𝑎9 𝑎10
Incorrect Code: 𝑒1 𝑎2 𝑎3 𝑎4 𝑎5 𝑎6 𝑎7 𝑎8 𝑎9 𝑎10
So the correct check digit is
(𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5 + 𝑎6 + 𝑎7 + 𝑎8 + 𝑎9 + 𝑎10 ) mod 9
and the incorrect check digit is
(𝑒1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5 + 𝑎6 + 𝑎7 + 𝑎8 + 𝑎9 + 𝑎10 ) mod 9
The error will NOT be caught if
(𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5 + 𝑎6 + 𝑎7 + 𝑎8 + 𝑎9 + 𝑎10 ) −
(𝑒1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5 + 𝑎6 + 𝑎7 + 𝑎8 + 𝑎9 + 𝑎10 ) is a multiple of 9.
This simplifies to
𝑎1 − 𝑒1 is a multiple of 9.
Multiples of 9 are ±{0, 9, 18, 27, 36, . . . }.
If 𝑎1 − 𝑒1 = 0, then 𝑎1 = 𝑒1 so we did not really make an error.
If 𝑎1 − 𝑒1 = ±9 (which can also be written as |𝑎1 − 𝑒1 | = 9), then 𝑎1 and
𝑒1 are digits that are separated by 9 units. Digits are 0, 1, 2, 3, 4, 5, 6, 7,
8, and 9. The only digits that are separated by 9 units are 0 and 9, so our
error was writing a 0 instead of a 9 or writing a 9 instead of a 0.
If |𝑎1 − 𝑒1 | = 18, then then 𝑎1 and 𝑒1 are digits that are separated by 18
units. The digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are never separated by
more than 9 units, so this and all higher multiples of 9 are impossible.
Therefore, all single-digit errors in the first digit other than substitutions of
0 for 9 or 9 for 0 would yield a different check digit and would be
caught.
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2 0 1 6
C h a p t e r
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P a g e |6
The same logic applies for each digit, so this scheme will miss errors
where |𝑎𝑖 − 𝑒𝑖 | = 9 and catch all other single-digit errors. This means
that all single-digit errors other than interchanging 0 and 9 are caught.
Example
Will the “sum of digits mod 9” check digit scheme that is used for money
orders detect all transposition of digits?
Let’s look at an adjacent transposition of the first two digits, 𝑎1 and 𝑎2 .
Correct Code: 𝑎1 𝑎2 𝑎3 𝑎4 𝑎5 𝑎6 𝑎7 𝑎8 𝑎9 𝑎10
Incorrect Code: 𝑎2 𝑎1 𝑎3 𝑎4 𝑎5 𝑎6 𝑎7 𝑎8 𝑎9 𝑎10
So the correct check digit is
(𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5 + 𝑎6 + 𝑎7 + 𝑎8 + 𝑎9 + 𝑎10 ) mod 9
and the incorrect check digit is
(𝑎2 + 𝑎1 + 𝑎3 + 𝑎4 + 𝑎5 + 𝑎6 + 𝑎7 + 𝑎8 + 𝑎9 + 𝑎10 ) mod 9
The error will NOT be caught if
(𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5 + 𝑎6 + 𝑎7 + 𝑎8 + 𝑎9 + 𝑎10 ) −
(𝑎2 + 𝑎1 + 𝑎3 + 𝑎4 + 𝑎5 + 𝑎6 + 𝑎7 + 𝑎8 + 𝑎9 + 𝑎10 ) is a multiple of 9.
This simplifies to
0 is a multiple of 9.
Because 0 is a multiple of 9, transposition of the first two digits will not be
caught. The same logic applies for the transposition of any two digits.
Therefore, transposition of any two digits in the “sum of the digits mod
9” check digit scheme will not be caught.
We could have worked this one without as much writing. The order of
addition does not change the sum, so the check digit will not be changed
(thus the error will not be caught) if we transpose digits in the “sum of
the digits mod 9” check digit scheme.
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P a g e |7
NUMBER mod 7
Airline Tickets, UPS packages, and Avis and National rental car
companies reserve the last digit as a check digit. The check digit is the
identification number mod 7.
Example
Is the check digit for the National rental car number 5007125 valid?
Note: The number in this problem includes the identification number with
the check digit appended to the end.
The “number mod 7” check digit scheme will detect all single-digit errors
with the exception of the pairs 0 and 7, 1 and 8, or 2 and 9.
The “number mod 7” check digit scheme will also detect transposition of
adjacent digits with the exception of the pairs 0 and 7, 1 and 8, or 2 and 9.
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C a r t e r ,
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C h a p t e r
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N u m b e r s
P a g e |8
Weighted Check Digit Schemes
Universal Product Codes (UPC)
 The blocks of digits indicate the category of the item, the
manufacturer, the item, and the check digit, respectively.
 Let 𝑆 = 𝟑𝑎1 + 𝑎2 + 𝟑𝑎3 + 𝑎4 + 𝟑𝑎5 + 𝑎6 + 𝟑𝑎7 + 𝑎8 + 𝟑𝑎9 + 𝑎10 + 𝟑𝑎11 + 𝑎12 .
The check digit, 𝑎12 is chosen so that S a multiple of 10.
(This is the equivalent to saying 𝑆 mod 10 ≡ 0).
 This scheme does not catch jump transpositions. However, it will
catch all single-position errors and most other errors.
Example
Check to see if the first code has a valid check digit.
Example
Determine the check digit for the second code.
Example
Why does the UPC scheme not detect jump transposition errors?
( c )
C a r t e r ,
E p s t e i n
a n d
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2 0 1 6
C h a p t e r
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N u m b e r s
P a g e |9
Example
Show that the UPC scheme detects all single-digit errors.
Let’s look at an error in the first digit, 𝑎1 .
Correct Code: 𝑎1 𝑎2 𝑎3 𝑎4 𝑎5 𝑎6 𝑎7 𝑎8 𝑎9 𝑎10 𝑎11 𝑎12
Incorrect Code: 𝑒1 𝑎2 𝑎3 𝑎4 𝑎5 𝑎6 𝑎7 𝑎8 𝑎9 𝑎10 𝑎11 𝑎12
So the correct check digit is
(3𝑎1 + 𝑎2 + 3𝑎3 + 𝑎4 + 3𝑎5 + 𝑎6 + 3𝑎7 + 𝑎8 + 3𝑎9 + 𝑎10 + 3𝑎11 + 𝑎12 ) mod 10
and the incorrect check digit is
(3𝑒1 + 𝑎2 + 3𝑎3 + 𝑎4 + 3𝑎5 + 𝑎6 + 3𝑎7 + 𝑎8 + 3𝑎9 + 𝑎10 + 3𝑎11 + 𝑎12 ) mod 10
The error will NOT be caught if
(3𝑎1 + 𝑎2 + 3𝑎3 + 𝑎4 + 3𝑎5 + 𝑎6 + 3𝑎7 + 𝑎8 + 3𝑎9 + 𝑎10 + 3𝑎11 + 𝑎12 ) −
(3𝑒1 + 𝑎2 + 3𝑎3 + 𝑎4 + 3𝑎5 + 𝑎6 + 3𝑎7 + 𝑎8 + 3𝑎9 + 𝑎10 + 3𝑎11 + 𝑎12 ) is a multiple of 10.
This simplifies to
3𝑎1 − 3𝑒1 is a multiple of 10,
which means 3(𝑎1 − 𝑒1 ) is a multiple of 10,
which means 𝑎1 − 𝑒1 is a multiple of
Multiples of
10
3
are ± {0,
10
3
,
20
3
, 10,
10
.
3
40
3
, . . . }.
If 𝑎1 − 𝑒1 = 0, then 𝑎1 = 𝑒1 so we did not really make an error.
10
If 𝑎1 − 𝑒1 = ± 3 , then 𝑎1 and 𝑒1 are digits that are separated by
10
3
. The
difference between digits is always an integer, so this and all other noninteger multiples of
10
3
are impossible.
If |𝑎1 − 𝑒1 | = 10, then then 𝑎1 and 𝑒1 are digits that are separated by 10
units. The digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are never separated by more
than 9 units, so this and all higher multiples of
10
3
are impossible.
Therefore, all single-digit errors in the first digit would be caught.
The same logic applies for all of the digits with a coefficient of 3 (the 3rd,
5th, 7th, 9th, and 11th digits), so this scheme will catch all single-digit
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N u m b e r s
P a g e | 10
errors in the odd-numbered positions. Therefore, we need to check the
even-numbered positions.
Let’s look at an error in the second digit, 𝑎2 .
Correct Code: 𝑎1 𝑎2 𝑎3 𝑎4 𝑎5 𝑎6 𝑎7 𝑎8 𝑎9 𝑎10 𝑎11 𝑎12
Incorrect Code: 𝑎1 𝑒2 𝑎3 𝑎4 𝑎5 𝑎6 𝑎7 𝑎8 𝑎9 𝑎10 𝑎11 𝑎12
So the correct check digit is
(3𝑎1 + 𝑎2 + 3𝑎3 + 𝑎4 + 3𝑎5 + 𝑎6 + 3𝑎7 + 𝑎8 + 3𝑎9 + 𝑎10 + 3𝑎11 + 𝑎12 ) mod 10
and the incorrect check digit is
(3𝑎1 + 𝑒2 + 3𝑎3 + 𝑎4 + 3𝑎5 + 𝑎6 + 3𝑎7 + 𝑎8 + 3𝑎9 + 𝑎10 + 3𝑎11 + 𝑎12 ) mod 10
The error will NOT be caught if
(3𝑎1 + 𝑎2 + 3𝑎3 + 𝑎4 + 3𝑎5 + 𝑎6 + 3𝑎7 + 𝑎8 + 3𝑎9 + 𝑎10 + 3𝑎11 + 𝑎12 ) −
(3𝑎1 + 𝑒2 + 3𝑎3 + 𝑎4 + 3𝑎5 + 𝑎6 + 3𝑎7 + 𝑎8 + 3𝑎9 + 𝑎10 + 3𝑎11 + 𝑎12 ) is a multiple of 10.
This simplifies to ________________________________
Multiples of 10 are _____________________________
If 𝑎2 − 𝑒2 = 0, ________________________________
If |𝑎2 − 𝑒2 | = 10, then then 𝑎2 and 𝑒2 are digits that are separated by 10
units. The digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are never separated by more
than 9 units, so this and all higher multiples of 10 are impossible.
Therefore, all single-digit errors in the second digit would be caught.
The same logic applies for all of the digits with a coefficient of 1 (the 4th,
6th, 8th, 10th, and 12th digits), so this scheme will catch all single-digit
errors in the even-numbered positions.
Therefore, the UPC scheme detects all single-digit errors.
( c )
C a r t e r ,
E p s t e i n
a n d
B o l l i n g e r
2 0 1 6
C h a p t e r
1 6 :
I d e n t i f i c a t i o n
N u m b e r s
P a g e | 11
Example
Will the UPC scheme detect adjacent transposition errors in the first and
second digits?
A transposition of the first and second digits would result in:
Correct Code: 𝑎1 𝑎2 𝑎3 𝑎4 𝑎5 𝑎6 𝑎7 𝑎8 𝑎9 𝑎10 𝑎11 𝑎12
Incorrect Code: 𝑎2 𝑎1 𝑎3 𝑎4 𝑎5 𝑎6 𝑎7 𝑎8 𝑎9 𝑎10 𝑎11 𝑎12
So the correct check digit is
(3𝑎1 + 𝑎2 + 3𝑎3 + 𝑎4 + 3𝑎5 + 𝑎6 + 3𝑎7 + 𝑎8 + 3𝑎9 + 𝑎10 + 3𝑎11 + 𝑎12 ) mod 10
and the incorrect check digit is
(3𝑎2 + 𝑎1 + 3𝑎3 + 𝑎4 + 3𝑎5 + 𝑎6 + 3𝑎7 + 𝑎8 + 3𝑎9 + 𝑎10 + 3𝑎11 + 𝑎12 ) mod 10
The error will not be caught if
(3𝑎1 + 𝑎2 + 3𝑎3 + 𝑎4 + 3𝑎5 + 𝑎6 + 3𝑎7 + 𝑎8 + 3𝑎9 + 𝑎10 + 3𝑎11 + 𝑎12 ) −
(3𝑎2 + 𝑎1 + 3𝑎3 + 𝑎4 + 3𝑎5 + 𝑎6 + 3𝑎7 + 𝑎8 + 3𝑎9 + 𝑎10 + 3𝑎11 + 𝑎12 ) is a multiple of 10.
This simplifies to
3𝑎1 + 𝑎2 − 3𝑎2 − 𝑎1 is a multiple of 10,
which means 2𝑎1 − 2𝑎2 = 2(𝑎1 − 𝑎2 ) is a multiple of 10,
which means 𝑎1 − 𝑎2 is a multiple of ______.
Multiples of ______ are _______________________________
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a n d
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2 0 1 6
C h a p t e r
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N u m b e r s
P a g e | 12
Banking
(Image from Nationwide.com)
 Let 𝑅 = 𝟕𝑎1 + 𝟑𝑎2 + 𝟗𝑎3 + 𝟕𝑎4 + 𝟑𝑎5 + 𝟗𝑎6 + 𝟕𝑎7 + 𝟑𝑎8 for the routing
number. The check-digit, 𝑎9 , equals the last digit of R.
 This scheme will catch all single-position errors and most other errors,
including most jump transpositions.
Example
Does this sample routing number have a valid check digit?
Example
The HR representative could not read a new doctor’s handwriting on her
direct deposit form. The HR representative could tell the routing number
was 314x77337. Can you determine the missing digit for the HR
representative?
x
R
0
1
2
3
4
5
6
7
8
9
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P a g e | 13
Credit Cards
Credit cards have 15-digit numbers with a check digit in position 16.
Let D = the sum of the digits in odd-numbered positions
Let E = the sum of the digits in even-numbered positions
(not including the check digit)
Let T = the number of digits in odd-numbered positions that are
larger than 4.
Let 𝐶 = 2𝐷 + 𝐸 + 𝑇 + 𝑎16
The check digit, 𝑎16 , is chosen so C is a multiple of 10.
This method detects single-position errors and 98% of other common
errors.
Example
The number 4128 0012 3456 7890 was listed on a credit card
advertisement. Is the check digit correct?
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E p s t e i n
a n d
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P a g e | 14
ISBN-10
The blocks of digits represent the language of the country in which the
book was published, the publisher, the book, and the check digit,
respectively.
Let 𝑉 = 𝟏𝟎𝑎1 + 𝟗𝑎2 + 𝟖𝑎3 + 𝟕𝑎4 + 𝟔𝑎5 + 𝟓𝑎6 + 𝟒𝑎7 + 𝟑𝑎8 + 𝟐𝑎9 + 𝑎10.
The check digit, 𝑎10 , is chosen so V is divisible by 11.
A remainder of 10 is possible when dividing by 11. If 𝑎10 needs to be
10, an X is used to represent 𝑎10 so it will take up only one space.
The scheme used for the 10-digit code detects all single-digit AND all
transposition errors!
Example
Is the check digit correct for the ISBN-10 listed above?
The scheme for ISBN-13 is similar to the UPC scheme and is described in
the summary at the end of the notes. Note the similarity between ISBN-10
and ISBN-13.
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a n d
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N u m b e r s
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Vehicle Identification codes
(Image from text page 581)
Read the description of the scheme for VINs from page 581 of the text and
determine if the check digit in this VIN is correct.
( c )
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E p s t e i n
a n d
B o l l i n g e r
2 0 1 6
C h a p t e r
1 6 :
I d e n t i f i c a t i o n
Example
Let the check digit 𝑎4 = (7𝑎1 + 2𝑎2 + 9𝑎3 ) mod 10
Will this catch all single-digit errors in the first digit?
N u m b e r s
P a g e | 16
( c )
C a r t e r ,
E p s t e i n
a n d
B o l l i n g e r
2 0 1 6
C h a p t e r
1 6 :
I d e n t i f i c a t i o n
Example
Let the check digit 𝑎4 = (7𝑎1 + 2𝑎2 + 9𝑎3 ) mod 10
Will this catch all single-digit errors in the second digit?
N u m b e r s
P a g e | 17
( c )
C a r t e r ,
E p s t e i n
a n d
B o l l i n g e r
2 0 1 6
C h a p t e r
1 6 :
I d e n t i f i c a t i o n
N u m b e r s
Example
Let’s keep the check digit 𝑎4 = (7𝑎1 + 2𝑎2 + 9𝑎3 ) mod 10
Will this catch all transposition errors in the first two digits?
P a g e | 18
( c )
C a r t e r ,
E p s t e i n
a n d
B o l l i n g e r
2 0 1 6
C h a p t e r
1 6 :
I d e n t i f i c a t i o n
Example
Let the check digit 𝑎3 = (3𝑎1 + 𝑎2 ) mod 11
Will this catch all single-digit errors in the first digit?
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C a r t e r ,
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a n d
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2 0 1 6
C h a p t e r
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I d e n t i f i c a t i o n
Example
Let’s keep the check digit 𝑎3 = (3𝑎1 + 𝑎2 ) mod 11
Will this catch all single-digit errors in the second digit?
N u m b e r s
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( c )
C a r t e r ,
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C h a p t e r
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I d e n t i f i c a t i o n
Example
Let’s keep the check digit 𝑎3 = (3𝑎1 + 𝑎2 ) mod 11
Will this catch all transposition errors?
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N u m b e r s
16.2 Zip Codes
(Image from text page 582)
Example
What does Texas A&M’s zip code of 77843-0001 tell us?
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16.3 Bar Codes
A system for representing data with only two symbols is a ___________
system.
A ______ _______ is a binary system that is made up of a series of dark
bars and light spaces that represent characters.
The ____________ code is used to encode ZIP+4 numbers. Each digit is
represented by _____ vertical bars (___ long and ____ short). The ZIP+4
is followed by a check digit determined by adding the first 9 digits and
choosing the check digit so the sum of all ten digits is a multiple of 10.
The bars for these ten digits are surrounded by a tall bar on each end
called guard bars to designate the beginning and end of the code.
Example
Determine the ZIP+4 displayed below and determine if the check digit is
correct.
(Image from text page 584)
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This special bar code is called a ________
_________ code. These can encode much
more information and are popular in print
media. The Cooking Light magazine from
September, 2011 had quite a few, including
this one.
16.4 Personal Data
Social Security
The first three digits tell the state where the application was filed.
Florida Driver’s License
(image from
http://restaurantandlodging.com/a-lacarte/new-law-permits-veterans-designationon-florida-driver-licen.html)




Take on the form XXXX-XXX-YY-DDD-N
XXXX-XXX is based on a coding of your name.
YY gives the last two digits of the year of your birth
DDD is coded from the month (m) and day (d) of your birth. These
three digits are
o Male: 40 (𝑚 − 1) + 𝑑
o Female: 40 (𝑚 − 1) + 𝑑 + 500
o This system uses 40 as a loose representation for the number of
days in a month.
 N is an overflow digit in case multiple people have the same number.
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Example
Are the digits YY-DDD correct for Joe Sample?
Example
Determine the Florida driver’s license digits YY-DDD for a female who
was born on July 18, 1942.
Example
What can you determine about a person who holds a Florida driver’s
license with the digits YY-DDD as 61-528?
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Example
What can you determine about a person who holds a Florida driver’s
license with the digits YY-DDD as 34-475?
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Summary of Check Digit Schemes
The check digit is the last digit unless noted otherwise.
Bank Routing Number
 The check digit, 𝑎9 , is the last digit of R where
𝑅 = 7𝑎1 + 3𝑎2 + 9𝑎3 + 7𝑎4 + 3𝑎5 + 9𝑎6 + 7𝑎7 + 3𝑎8
Codabar Scheme - Credit Cards, Libraries, Blood Banks, South Dakota
Driver’s License
 The smallest nonnegative integer, 𝑎16 , such that 𝐶 is a multiple of 10
where
𝐶 = 2𝐷 + 𝐸 + 𝑇 + 𝑎16 and
𝐷 = the sum of the digits in odd-numbered positions
𝐸 = the sum of the digits in even-numbered positions
(not including the check digit)
𝑇 = the number of digits in odd-numbered positions that are larger
than 4
ISBN-10
 The smallest nonnegative integer, 𝑎10 , such that 𝑉 is a multiple of 11
where
𝑉 = 10𝑎1 + 9𝑎2 + 8𝑎3 + 7𝑎4 + 6𝑎5 + 5𝑎6 + 4𝑎7 + 3𝑎8 + 2𝑎9 + 𝑎10 .
 The letter X is used to represent the integer 10.
ISBN-13 and EAN (European Article Number)
 The smallest nonnegative integer, 𝑎13 , such that 𝑆 is a multiple of 10
where
𝑆 = 𝑎1 + 3𝑎2 + 𝑎3 + 3𝑎4 + 𝑎5 + 3𝑎6 + 𝑎7 + 3𝑎8 + 𝑎9 + 3𝑎10 + 𝑎11 + 3𝑎12 + 𝑎13
Money Order from US Postal Service
 The check digit, 𝑎11 , is the sum of the first ten digits mod 9
Rental Cars (Avis and National) and Airline Tickets
 The number formed by the digits prior to the check digit mod 7.
Note that this does NOT use the sum of the digits.
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Travelers Cheques (American Express and Visa) and Euro Bank Notes
 The smallest nonnegative integer such that the sum of the digits
(including the check digit) is a multiple of 9.
UPC (Universal Product Code)
 The smallest nonnegative integer, 𝑎12 , such that S is a multiple of 10
where
𝑆 = 3𝑎1 + 𝑎2 + 3𝑎3 + 𝑎4 + 3𝑎5 + 𝑎6 + 3𝑎7 + 𝑎8 + 3𝑎9 + 𝑎10 + 3𝑎11 + 𝑎12
ZIP+4
 The smallest nonnegative integer such that the sum of the digits
(including the check digit) is a multiple of 10.