Universal product codes (UPC) are the familiar bar codes that identify products so that they can be automatically priced at the checkout counter. A UPC is a 12-digit code in which the first digit characterizes the type of product (0 identifies an ordinary grocery item, 2 is an item sold by weight. 3 is a medical item, 4 is a special item, 5 is a coupon, and 6 and 7 are items not sold in retail stores). The next five digits identify the manufacturer, the next five digits identify the product, and the last digit is a check digit. (All UPC codes have a check digit. It is always present on the bar code, but it may not appear in the printed version.) For example, the UPC for a package of 10 Ortega taco shells is 0-54400-00800-5. The first zero identifies this as an ordinary grocery item, the next five digits 54400 identity the manufacturer Nabisco Foods, and the next five digits 00800 identify the product as a package of 10 Ortega taco shells.
The check digit is computed as follows. First compute s, where s is 3 times the sum of every other number starting with the first plus the sum of every other number, except the check digit, starting with the second. The check digit is the number c, between 0 and 9 satisfying (c + s) mod 10 = 0. For the code on the package of taco shells, we would have
s = 3(0 + 4 − 4 + 0 + 0 + 84-0)4-5 + 4-1-0 + 0 + 0 = 45.
Since (5 + 45) mod 10 = 0, the check digit is 5.
Find the check digit for the UPC whose first 11 digits are 3-41280-21414.
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