  # PreTeXt Sample Book: Abstract Algebra (SAMPLE ONLY)

### 1.UPC Symbols.

Universal Product Code (UPC) symbols are found on most products in grocery and retail stores. The UPC symbol is a 12-digit code identifying the manufacturer of a product and the product itself (Figure 3.6.1). The first 11 digits contain information about the product; the twelfth digit is used for error detection. If $$d_1 d_2 \cdots d_{12}$$ is a valid UPC number, then
\begin{equation*} 3 \cdot d_1 + 1 \cdot d_2 + 3 \cdot d_3 + \cdots + 3 \cdot d_{11} + 1 \cdot d_{12} \equiv 0 \pmod{10}\text{.} \end{equation*}
1. Show that the UPC number 0-50000-30042-6, which appears in Figure 3.6.1, is a valid UPC number.
2. Show that the number 0-50000-30043-6 is not a valid UPC number.
3. Write a formula to calculate the check digit, $$d_{12}\text{,}$$ in the UPC number.
4. The UPC error detection scheme can detect most transposition errors; that is, it can determine if two digits have been interchanged. Show that the transposition error 0-05000-30042-6 is not detected. Find a transposition error that is detected. Can you find a general rule for the types of transposition errors that can be detected?
5. Write a program that will determine whether or not a UPC number is valid.

### 2.

It is often useful to use an inner product notation for this type of error detection scheme; hence, we will use the notion
\begin{equation*} (d_1, d_2, \ldots, d_k ) \cdot (w_1, w_2, \ldots, w_k ) \equiv 0 \pmod{ n } \end{equation*}
to mean
\begin{equation*} d_1 w_1 + d_2 w_2 + \cdots + d_k w_k \equiv 0 \pmod{ n}\text{.} \end{equation*}
Suppose that $$(d_1, d_2, \ldots, d_k ) \cdot (w_1, w_2, \ldots, w_k ) \equiv 0 \pmod{ n}$$ is an error detection scheme for the $$k$$-digit identification number $$d_1 d_2 \cdots d_k\text{,}$$ where $$0 \leq d_i \lt n\text{.}$$ Prove that all single-digit errors are detected if and only if $$\gcd( w_i, n ) = 1$$ for $$1 \leq i \leq k\text{.}$$

### 3.

Let $$(d_1, d_2, \ldots, d_k ) \cdot (w_1, w_2, \ldots, w_k ) \equiv 0 \pmod{ n}$$ be an error detection scheme for the $$k$$-digit identification number $$d_1 d_2 \cdots d_k\text{,}$$ where $$0 \leq d_i \lt n\text{.}$$ Prove that all transposition errors of two digits $$d_i$$ and $$d_j$$ are detected if and only if $$\gcd( w_i - w_j, n ) = 1$$ for $$i$$ and $$j$$ between 1 and $$k\text{.}$$

### 4.ISBN Codes.

Every book has an International Standard Book Number (ISBN) code. This is a 10-digit code indicating the book's publisher and title. The tenth digit is a check digit satisfying
\begin{equation*} (d_1, d_2, \ldots, d_{10} ) \cdot (10, 9, \ldots, 1 ) \equiv 0 \pmod{11}\text{.} \end{equation*}
One problem is that $$d_{10}$$ might have to be a 10 to make the inner product zero; in this case, 11 digits would be needed to make this scheme work. Therefore, the character X is used for the eleventh digit. So ISBN 3-540-96035-X is a valid ISBN code.
1. Is ISBN 0-534-91500-0 a valid ISBN code? What about ISBN 0-534-91700-0 and ISBN 0-534-19500-0?
2. Does this method detect all single-digit errors? What about all transposition errors?
3. How many different ISBN codes are there?
4. Write a computer program that will calculate the check digit for the first nine digits of an ISBN code.
5. A publisher has houses in Germany and the United States. Its German prefix is 3-540. If its United States prefix will be 0-abc, find abc such that the rest of the ISBN code will be the same for a book printed in Germany and in the United States. Under the ISBN coding method the first digit identifies the language; German is 3 and English is 0. The next group of numbers identifies the publisher, and the last group identifies the specific book.