I need help the part "show that the check digit of an ISBN-13 can always detect a single error and "show that there are transpositions of two digits that are not detected by an ISBN-13

ISBN 978-0-07-7431440 With more people doing more things with more objects in more places than ever before, even more data is being generated every day. And while the sheer amount and scope of all this data makes it very difficult to analyze, it also makes it that much harder to guarantee the validity of that data. For the most part, data entry relies on human operators, which are prone to human error. And even when it is (less error-prone but still not perfect) computers only talking to other computers, the simple act of moving data from one place to another involves physical transmission which contains its own error rates. Items are produced and stocked according to Universal Product Codes, shipped from place to place in containers marked with Routing Numbers, and purchased using the Account Number of a bank or credit card. All of which have to be verified quickly and often. One way of doing this is to use a Check Digit protocol that compares the overall data to a single digit to make sure they are in agreement. (In fact, our own cells use something similar to fight against mutation \"errors" in our DNA.) While this works well enough when the error rate is small, systems prone to more errors or intentional spoofing may require more complicated methods. For this Project you will investigate the International Standard Book Number and its check digit protocol. You will have to find the ISBN number for the book used in the class, which means you may have to read the syllabus. You may also find it useful to read up on ISBN and other check digit methods in general. If you do any external research be sure to properly cite your sources. If you work with another student(s), please include their name(s) in the write-up as well. If you use any electronic resources to help you find the ISBN or perform calculations of the check digit, be sure to name them in the report. You should also perform the check digit calculations yourself, by hand, just to check. The check digit a1; for an ISBN-13 with initial digits 3132...312 is determined by the congruence (a1 + a3 + + 313) + 3(a2 + a4 + + 312) E 0 (mod 10). Find the ISBN13 for the text used in class, and (hopefully!) confirm that it is valid. Then show that the check digit of an ISBN-13 can always detect a single error. Finally, show that there are transpositions of two digits that are not detected by an ISBN-13