Check Digits
What?
Division Schemes
Weighted Schemes
Examples
Spring 2015
Mathematics in
Management Science
Check Digits
An additional digit attached to number
for purposes of error & fraud detection
and correction when copying or
transmitting the number.
Mathematical calculations (schemes)
used on the digits of the original
number to compute the check digit.
Check Digits
Check digit is calculated from the rest
of the number and transmitted along
with the number.
When an error occurs, a recalculation
of the check digit (usually) won’t match.
When the recalculated value doesn’t
match the supplied check digit, an error
has occurred.
Check Digits
Start with pre-id number. Use this to
calculate check digit. Get id number.
Check digit is extra digit appended to
pre-id number for purposes of
detecting/correcting errors (or fraud)
when the complete id number is copied
or transmitted.
Check Digits Examples
Calculation of a check digit typically
uses modular arithmetic.
You are not expected to memorize
rules in all the examples.
You are expected to be able to
calculate a check digit given a
description of the procedure.
USPS Money Order
U.S. Postal Service money order with
id number 6302438384, check digit 5.
The check digit in this case is the calculated remainder
after dividing the sum of the first ten digits by 9.
USPS Money Order
Length of ID: 10 + check digit
a1a2a3a4a5a6a7a8a9a10 c
Calculating check digit:
c = a1a2a3a4a5a6a7a8a9a10 mod 9 =
(a1+a2+a3+a4+a5+a6+a7+a8+a9+a10 ) mod 9
Errors detected/not detected:
Single digit errors except replacing 0 with 9
or vice versa. Won’t detect transpositions.
Example
Find check digit for money order id
2755943610c.
c = (2+7+5+5+9+4+3+6+1+0) mod 9
= 42 mod 9
=6
So, complete id number is
27559436106.
Example
Find check digit for money order id
5602419260c.
c = (5+6+0+2+4+1+9+2+6+0) mod 9
= 35 mod 9
=8
So, complete id number is
5602419268.
American Express Cheques
Another division by 9 scheme.
Id no has form a1a2a3a4a5a6a7a8a9 c ;
The check digit c is chosen so that
a1 +a2+a3+a4+a5+a6+a7+a8+a9 +c
is evenly divisible by 9.
Same error detection as USPS money
orders.
Example
Find check digit for AmEx check:
603780104c.
Need
(6+0 +3+7+8+0+1+0+4) mod 9+c
= 29 mod 9+c
= 2+c to equal either 0 or 9.
So, c = 7 and complete ID is
6037801047.
Direct vs Indirect Methods
USPS money order check digit is computed
via direct calculation: c is remainder after
division by 9.
AmExpress cheques check digit is computed
via indirect calculation: c is chosen so that
sum is evenly divisible by 9.
Direct methods easier to compute, but harder
to verify.
Indirect methods harder to compute, but
easier to verify.
Detecting Errors
Direct Method: Recalculate check
digit and compare it to one supplied.
Indirect Method: Use supplied check
digit and see if it satisfies condition.
Other Schemes
Can be “direct” or “indirect”.
Can use division by 7 or 10 or 11 or 13.
Can introduce “weights”:
UPC, BIN, ISBN, codabar
Do not need to memorize all these!
But, shud be able to calculate a check
digit given description of scheme.
Indirect Div by 10 Example
This a division by 10 scheme.
11 digits + check digit
Check digit c chosen so that sum of all
digits is divisible by 10.
Id no of form a1a2a3a4a5a6a7a8a9a10 c
Check digit c is chosen so that
a1 +a2+a3+a4+a5+a6+a7+a8+a9 +a10+c
is divisible by 10.
Universal Product Code
A bar code and identification number
that is used on most retail items.
By using a weighted scheme in the
calculation of the check digit, the UPC
number achieves greater error
detection—up to 100% of all single-digit
errors and most other types of errors.
Example
Consider the number 0 38000 00127 7
found on bottom of box of cornflakes.
First digit identifies broad category of
goods.
Next five digits identify manufacturer.
Next five digits identify product.
The last is a check digit.
Universal Product Codes
This is a weighted div by 10 scheme.
Pre-id no: a1a2a3a4a5a6a7a8a9a10a11
Id no: pre-id no + check digit
Check digit c chosen so that weighted
sum
3a1+a2+3a3+a4+3a5+a6+3a7+a8+3a9+a10+3a11+c
is divisible by 10.
Weights are 3,1 (repeated over & over).
Example
Look at UPC no 0 38000 00127 7.
Weights are
3 13131 31313 1.
Weighted sum of all these is
0+3+3*8+3*1+2+3*7+ 7=27+5+28=60
Which is divisible by 10.
Example
Find check digits for the UPC pre-id
number
0 31200 00678.
Weights are 3 13131 31313 1.
Weighted sum of all these is
0+3+3+2+3*6+7+3*8+c=57+c.
Want this to be divisible by 10.
So, c=3. Then UPC id no is
0 31200 00678 3.
Example
Find check digits for the UPC pre-id
number
0 74101 75074.
Weights are 3 13131 31313 1.
Weighted sum of all these is
7+3*4+1+1+3*7+5+7+3*4+c=66+c.
Want this to be divisible by 10.
So, c=4. Then UPC id no is
0 74101 75074 4.
Example
Is 0 38000 00097 2 a valid UPC no?
3 13131 31313 1 (weights).
Weighted sum of all these is
3+3*8+9+3*7+2 = 59.
Since this is not divisible by 10, above
number is not a valid UPC number.
Correcting Errors
Suppose get a UPC number
0-46000-x2121-6
where the 7th digit x could not be read.
So, this is, of course, an error.
Can we fix this? Can we determine
what digit the x should be?
Yes, since it’s just one digit.
Have a wanna be UPC number
0-46000-x2121-6.
Need x to have prop that wtd sum is
divisible by 10:
0 46 000 x21 21 6
3 13 131 313 13 1
Sum is
4+3*6+3x+2+3+2+3+6=38+3x.
This div by 10 iff 8+3x is div by 10.
Want x st 8+3x is divisible by 10.
Solving 3x+8=0 mod 10
A number N is divisible by 10 iff
N mod 10 = 0, or, N ≡ 0 mod 10 .
Two numbers M and N satisfy
M≡N mod 10 iff M-N is divisible by 10.
So, 1≡11 mod 10, 2≡-8 mod 10, etc.
We want 3x ≡ -8 mod 10, but
-8 ≡ 2 ≡ 12 mod 10,
so want 3x ≡ 12 mod 10; get x=4 .
Solving 3x+8=0 mod 10
Another (better?) approach:
We want 3x ≡ -8 ≡ 2 mod 10.
Notice that 3 • 7 = 21 ≡ 1 mod 10.
So, doing mod 10 arithmetic, we
multiply both sides of 3x = 2 by 7 to
get
x ≡ 7 • 3x ≡ 7 • 2 ≡ 4 mod 10.
Again we find that x=4 .
Modular Arithmetic
The set of integers “mod p” are
0, 1, 2, 3, …, p-1 .
Just all remainders after division by p.
When p=5 get 0, 1, 2, 3, 4.
We add these by computing “mod p”.
When p=5, 3+4 ≡ 2 mod 5.
Do something similar for multiplication.
Modular Multiplication
The integers mod 5: 0, 1, 2, 3, 4 .
All possible remainders after div by 5.
Multiplication Table.
X
1
2
3
4
1
1
2
3
4
2
2
4
1
3
3
3
1
4
2
4
4
3
2
1