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Chapter 3: Problem 71

Determine which ISBNs (see Exercise 65) have correct check digits. $$ 978-0-684-87018-0 $$

Short Answer

Expert verified
The ISBN-13 number 978-0-684-87018-0 is not valid.

Step by step solution

01

Prepare the ISBN for calculation

Remove the hyphens from the ISBN number and input each digit into an array for processing. The given ISBN number is 978-0-684-87018-0, so the resulting array will be [9,7,8,0,6,8,4,8,7,0,1,8,0].
02

Calculate the ISBN check digit

Starting from the left, alternate between multiplying each digit by 1 or 3, then sum all these products. The formula is as thus: \( sum = (9*1+7*3+8*1+0*3+6*1+8*3+4*1+8*3+7*1+0*3+1*1+8*3+0*1) = 108 \).
03

Validate the Check Digit

Get the modulus 10 of the sum calculated in step 2. If the result is equal to 0, then the number is a valid ISBN-13. In this case, \( 108 mod 10 = 8 \), since the result is not 0, the ISBN-13 is not valid.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discrete Mathematics
Discrete mathematics is a branch of mathematics that deals with objects that can assume only distinct, separated values. This means it focuses on study of things that can be counted. When we look at ISBN check digits, we're delving into a special realm of discrete mathematics where patterns and sequences play crucial roles.
Modular Arithmetic
Modular arithmetic is a system of arithmetic for integers, where numbers wrap around upon reaching a certain value, known as the modulus. Commonly referred to as 'clock arithmetic,' it's the mathematical foundation of many digital systems, including the calculation of ISBN check digits. When the textbook solution calculated the sum and then used mod 10, it was applying modular arithmetic. This means that after reaching 10, our number wraps around to zero. It's essential for ensuring the last digit of an ISBN is accurately calculated and for error detection in various digital communications.
Error Detection Codes
Error detection codes play a vital role in ensuring data integrity during transmission or storage. ISBN utilizes a special error detection method where the check digit ensures the rest of the number sequence is correct. This mechanism allows for the detection of single-digit errors and most transposition errors. By using a simple mathematical formula and modular arithmetic, error detection codes can quickly verify the accuracy of data such as an ISBN number.

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Most popular questions from this chapter

Use the following definitions. Let \(U\) be a universal set and let \(X \subseteq U\). Define $$ C_{X}(x)=\left\\{\begin{array}{ll} 1 & \text { if } x \in X \\ 0 & \text { if } x \notin X . \end{array}\right. $$ We call \(C_{X}\) the characteristic function of \(X(\) in \(U) .\) (A look ahead at the next Problem-Solving Corner may help in understanding the following exercises.) Prove that the function \(f\) from \(\mathcal{P}(U)\) to the set of characteristic functions in \(U\) defined by $$ f(X)=C_{X} $$ is one-to-one and onto.

For the sequence \(r\) defined by $$r_{n}=3 \cdot 2^{n}-4 \cdot 5^{n}, \quad n \geq 0$$. Find \(r_{1}\)

Use the following definitions. Let \(U\) be a universal set and let \(X \subseteq U\). Define $$ C_{X}(x)=\left\\{\begin{array}{ll} 1 & \text { if } x \in X \\ 0 & \text { if } x \notin X . \end{array}\right. $$ We call \(C_{X}\) the characteristic function of \(X(\) in \(U) .\) (A look ahead at the next Problem-Solving Corner may help in understanding the following exercises.) Prove that \(C_{X-Y}(x)=C_{X}(x)\left[1-C_{Y}(x)\right]\) for all \(x \in U\).

Find the months with Friday the 13 th in 2040 .

Prove that for all real numbers \(x\) and integers \(n,\lceil x\rceil=\) \(n\) if and only if there exists \(\varepsilon, 0 \leq \varepsilon<1\), such that \(x+\varepsilon=n\)
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