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posted by  StrickerXY on 10/3/2008 12:17:29 PM  |  status: Live  

number theory2 modulo

Course Textbook Chapter Problem
Discrete Math Discrete mathematics and its applications, sixth edition, kenneth rosen N/A N/A
Question Details:

Another application using modulo…The EAN-13 barcode was implemented by the International

Article Numbering Association (EAN) in Europe and is used in all media types, not just books like the

ISBN. An EAN-13 barcode is divided into four areas: 1) The number system, 2) The manufacturer

code, 3) the product code, and 4) the check digit. [See example below, the check digit is the last digit

on the right-hand side below the barcode, in this case ‘5’.]

The steps for calculating the check digit are as follows:

_ Starting with the number system, the manufacturer code, and the product code (altogether

they are 12 digits), consider the right-most digit of the number to be in an "odd" position, and

assign odd/even to each character moving from right to left.

_ Sum the digits in all odd positions, and multiply the result by 3.

_ Sum the digits in all even positions.

_ Sum the totals calculated in steps 2 and 3.

_ The check digit is the number which, when added to the totals calculated in step 4, result in a

number evenly divisible by 10. If the sum calculated in step 4 is evenly disivisible by 10, the

check digit is "0" (not 10).

a. Show the check digit in the example above is correct.

b. Find the check digit for the 359671001729_.
 
 
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posted by Nischal.S. on 10/3/2008 1:09:06 PM  |  status: Live
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StrickerXY's comment:
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Response Details:
FACT: A number is divisible by 10 if its LAST digit is 0--->all we care is whether the last digit is 0 or not irrespecitve of other digits;
Now, if the sum obtained in step 4 is divisible by 10.i,e the last digit of the sum obtained in step 4 is 0;
 And suppose we append the check digit to be 0;
i.e we do not change the sum obtained in step 4; ∴The last digit is still zero;
Hence, the check digit when added to the totals obtained in step 4 is divisible by 10;
∴Check digit in the given example is correct

b) Given 359671001729_
∴digits in odd position={9,7,0,1,6,5} and their sum of digits= 28;sum*3 = 28*3=84
similarly digits in even position ={2,1,0,7,9,3} and their sum = 22;
Total sum = 84+22 = 106;

check digit = 4;
bcoz if we add the check digit  4 to the total 106 we get 110 which is evenly divisible by 4
Nischal. S.
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