The availability of inexpensive, fast, reliable scanning devices and computers has made the appendage of a check digit to identification numbers a standard practice. Indeed, one finds check digits appended to identification numbers on airline tickets, credit cards, money orders, bank accounts, checking accounts, library books, grocery items, traveler's checks, driver's licenses, passports, rental cars, chemicals, blood bank items, photofinishing envelopes, UPS packages, express mail, bar coded mail and books. Details about many of these are contained in [2],[3],[4] and [5].

Typical of the genre is the Universal Product Code (Figure 1) found on grocery items. A UPC identification number consists of twelve digits, each of which can be from 0 to 9. The first six digits identify the country and the manufacturer, the next five the product, the last is the check digit.(For many items, the check digit is not printed, but it is always bar coded.) The check digit a 12 for the UPC number a 1 a 2•·· a 11 is chosen to satisfy the condition