The Universal Product Codes (UPC) are 12-digits identifiers where the 11 first digits are manufacturer/product information and the last digit is a check digit. An UPC
c=(a1a2,...a12) is valid if
3a1 + a2 + 3a3 + a4 + ⋯ + 3a11 + a12 = 0 (mod10)
a. A single digit error on a UPC occurs when exactly one of the digits a1,…,a11 changes to another digit. Show that the check digit always detects single digit errors. b. Consider the product with UPC c=(78Q162001511). Determine the digit Q so the UPC is valid. What is the product determined by c? c. A transposition error on a UPC occurs when two of the digits a1,⋯,a11 are interchanged. Show that the checking digit detects the transposition error on c1=(78Q126001511) and does not detect an error on c2=(78Q612001511). (You must use the digit Q found in part a.) d. Explain what type of transposition errors on an UPC will be detected and what won't (in general) by the check digit.