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Identifying and analyzing the strings of digits seen on everyday products What are Check Digits? Modular Arithmetic and Divisibility Rules Proof Where we see Check Digits 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 EX: 66 Ξ 6 mod 10 10|(66-6) , where 10|60 or, 66/10 = 6 remainder 6 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 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 The first part is divisible by 9, so the second part can be substituted for the entire number 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 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 Using the Check Digit ◦ 3(a1+a3+...+a11)+(a2+a4+...+a12)Ξ 0 mod 10 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 Assuming the digits are "a1,a2... a13 , where a13 is the check digit a13 = a1+a3 +... +a11 +3*(a2+a4 +... +a12 ) mod(10) 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 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. 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.