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The EAN-13 barcode (European Article Number) is a 13-digit barcode standard used internationally to identify products. The last digit of the EAN-13 barcode is a check digit, which is used to ensure that the barcode is correctly composed. Here's a detailed description of how the check digit is calculated:

Structure of EAN-13

An EAN-13 barcode consists of 13 digits:

'The first 12 digits are the data digits.

'The 13th digit is the check digit.

Calculation of the Check Digit

The check digit is calculated using a specific algorithm based on the first 12 digits. The steps are as follows:

1.Start with the first 12 digits: Let's denote the 12 digits as d1,d2,d3,...,d12d_1, d_2, d_3, ..., d_{12}d1,d2,d3,...,d12.

2.Sum the digits in the odd positions (1, 3, 5, ..., 11):

Sum1=d1+d3+d5+d7+d9+d11\text{Sum1} = d_1 + d_3 + d_5 + d_7 + d_9 + d_{11}Sum1=d1+d3+d5+d7+d9+d11

3.Sum the digits in the even positions (2, 4, 6, ..., 12):

Sum2=d2+d4+d6+d8+d10+d12\text{Sum2} = d_2 + d_4 + d_6 + d_8 + d_{10} + d_{12}Sum2=d2+d4+d6+d8+d10+d12

4.Multiply the sum of the even-positioned digits by 3:

Sum2×3\text{Sum2} \times 3Sum2×3

5.Add the result from step 4 to the sum of the odd-positioned digits:

Total Sum=Sum1+(Sum2×3)\text{Total Sum} = \text{Sum1} + (\text{Sum2} \times 3)Total Sum=Sum1+(Sum2×3)

6.Calculate the check digit: The check digit is the smallest number which, when added to the total sum, results in a number that is a multiple of 10. This can be found using the formula:

Check Digit=(10 (Total Summod 10))mod 10\text{Check Digit} = (10 - (\text{Total Sum} \mod 10)) \mod 10Check Digit=(10 (Total Summod10))mod10

Here, \mod represents the modulo operation, which gives the remainder of the division.

Example Calculation

Suppose we have the first 12 digits of an EAN-13 barcode as: 400638133393.

1.Odd-positioned digits sum: 4+0+6+8+3+3=244 + 0 + 6 + 8 + 3 + 3 = 244+0+6+8+3+3=24

2.Even-positioned digits sum: 0+6+3+1+3+9=220 + 6 + 3 + 1 + 3 + 9 = 220+6+3+1+3+9=22

3.Multiply the even sum by 3: 22×3=6622 \times 3 = 6622×3=66

4.Total Sum: 24+66=9024 + 66 = 9024+66=90

5.Calculate the check digit:

Check Digit=(10 (90mod 10))mod 10=(10 0)mod 10=10mod 10=0\text{Check Digit} = (10 - (90 \mod 10)) \mod 10 = (10 - 0) \mod 10 = 10 \mod 10 = 0Check Digit=(10 (90mod10))mod10=(10 0)mod10=10mod10=0

So, the check digit for the EAN-13 code 400638133393 is 0, making the full EAN-13 barcode 4006381333930.

Summary

The check digit in an EAN-13 barcode is crucial for error detection. It ensures that the barcode is correctly composed and scanned by verifying that the digits conform to the algorithm. This helps in reducing errors in product identification and enhances the reliability of the barcode system.

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