Virtually every item sold in a store has a 12-digit UPC barcode which is scanned...
Virtually every item sold in a store has a 12-digit UPC barcode which is scanned at the checkout counter. The first 11 digits of a UPC number d1d2d3 ? ? ? d11d12 identify the manufacturer and product. The last digit d12 is a check digit which is chosen so that 3d1+d2+3d3+d4+3d5+d6+3d7+d8+3d9+d10+3d11+d12 =0 pmod10q.
If the congruence does not hold, an error has been made and the item must be scanned again, or the UPC code enteblack by hand. Which of the following UPC numbers were scanned incorrectly?