Transcript ISBN Check Digits

Identifying and analyzing the strings of digits
seen on everyday products
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What are Check Digits?
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Modular Arithmetic and Divisibility Rules
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Proof
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Where we see Check Digits
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Whenever you see a long list of numbers on a
product code, around 10 digits long, there is a
good chance the last digit is a check digit
Check digits exist to detect error through use of
basic modular arithmetic based on the
proceeding numbers in the code and a specific
divisibility factor
The arithmetic used to detect error can vary from
simple to complex, depending on the product
x Ξ y mod m , if and only if, m|(x-y)
Could be viewed as x/m = k with
remainder y
 Ξ means congruent
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EX: 66 Ξ 6 mod 10
10|(66-6) , where 10|60
or, 66/10 = 6 remainder 6
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3 – The sum of the digits are divisible by 3
7 – 3x + L , where L is the last digit, and x is
the numbers to the left of L
9 – The sum of the digits is divisible by 9
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Given : n is a whole number, with digits,
a1,a2,...,a5 (for example), 54321
This can be written as,
54321
=(10,000*5)+(1,000*4)+(100*3)+(10*2)+(1*1),
=(9,999*5+5)+(999*4+4)+(99*3+3)+(9*2+2)+1,
=9(1,111*5+111*4+11*3+2)+5+4+3+2+1
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The first part is divisible by 9, so the second
part can be substituted for the entire number
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Starting with the digits given, take the sum of the
digits until there is only one digit left.
The negative of the digit mod 9 will be the check
digit.
Letters are used in
Euro Bills,
use the following key.
A=11, B=12, C=13,
…, Z=36
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12 Digit Number
(a,b1,b2,b3,b4,b5,c1,c2,c3,c4,c5,d)
◦ a: category of goods,
◦ bi: manufacturer’s code
◦ ci: code for the product
◦ d: check digit
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Using the Check Digit
◦ 3(a1+a3+...+a11)+(a2+a4+...+a12)Ξ 0 mod 10
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Without using the Check Digit
◦ 3(a1+a3+...+a11)+(a2+a4+...+a10)Ξ (10-a12 )mod 10
Double every other number starting from
an-1 -> 2 an-1
and then moving to the left from there.
Then take the sum of the digits.
Multiply that sum by 9.
The last digit of
the product is
the check digit
Has the property
a1,a2... a10
Such that,
10a1+9a2+8a3+7a4+6a5+5a6+4a7+3a8+2a9+1a10 is
divisible by 11
a10 is the check digit in this situation
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Assuming the digits are "a1,a2... a13 , where
a13 is the check digit
a13 =
a1+a3 +... +a11 +3*(a2+a4 +... +a12 ) mod(10)
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Check Digits can be used to a multitude of
different products
All Check Digits rely on Modular Arithmetic
There are endless possibilities of generating
Check Digits
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Special thanks to our advisor for taking time
out of her day to help us learn and teach
others about the existence and ideas behind
Check Digits.
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Bleyer, Craig, and Randi Rossignol, eds. For All
Practical Purposes: Mathematical Literacy in
Today's World. 6th ed. New York: W.H. Freeman,
2000. Print.
Gärtner, Matthias. "Check Digits." Check Digits.
10 Sept. 2006. Web. 27 Apr. 2012.
<http://www.rtner.de/software/CheckDigits.htm
l>.
Velleman, Daniel J. How to Prove It: A Structured
Approach. 2nd ed. Cambridge: Cambridge UP,
2006. Print.