# Determine whether each of the ISBNs below is correct

• Math100
In summary: But I think the code is so constructed that the ten digit number works too. You only have to shift the number a bit. So we can check, whether the shifting is done correctly, or whether the number is false.
Math100
Homework Statement
The International Standard Book Number (ISBN) used in many libraries consists of nine digits ## a_{1}a_{2}...a_{9} ## followed by a tenth check digit ## a_{10} ##, which satisfies
## a_{10}\equiv \sum^{9}_{k=1} ka_{k}\pmod {11} ##.
Determine whether each of the ISBNs below is correct:
(a) 0-07-232569-0 (United States).
(b) 91-7643-497-5 (Sweden).
(c) 1-56947-303-10 (England).
Relevant Equations
None.
(a)
Consider the ISBN ## 0-07-232569-0 ## from the United States.
Observe that
\begin{align*}
&\sum^{9}_{k=1} ka_{k}\pmod {11}\\
&\equiv (a_{1}+2a_{2}+\dotsb +9a_{9})\pmod {11}\\
&\equiv (0+2\cdot 0+3\cdot 7+4\cdot 2+5\cdot 3+6\cdot 2+7\cdot 5+8\cdot 6+9\cdot 9)\pmod {11}\\
&\equiv (21+8+15+12+35+48+81)\pmod {11}\\
&\equiv 220\pmod {11}\\
&\equiv 0\pmod {11}.\\
\end{align*}
Thus ## \sum^{9}_{k=1} ka_{k}\pmod {11}\equiv a_{10} ##.
Therefore, the ISBN ## 0-07-232569-0 ## is correct.

(b)
Consider the ISBN ## 91-7643-497-5 ## from Sweden.
Observe that
\begin{align*}
&\sum^{9}_{k=1} ka_{k}\pmod {11}\\
&\equiv (a_{1}+2a_{2}+\dotsb +9a_{9})\pmod {11}\\
&\equiv (9+2\cdot 1+3\cdot 7+4\cdot 6+5\cdot 4+6\cdot 3+7\cdot 4+8\cdot 9+9\cdot 7)\pmod {11}\\
&\equiv (9+2+21+24+20+18+28+72+63)\pmod {11}\\
&\equiv 257\pmod {11}\\
&\equiv 4\pmod {11}.\\
\end{align*}
Thus ## \sum^{9}_{k=1} ka_{k}\pmod {11}\not\equiv a_{10} ## because ## a_{10}=5 ##.
Therefore, the ISBN ## 91-7643-497-5 ## is not correct.

(c)
Consider the ISBN ## 1-56947-303-10 ## from England.
Observe that
\begin{align*}
&\sum^{9}_{k=1} ka_{k}\pmod {11}\\
&\equiv (a_{1}+2a_{2}+\dotsb +9a_{9})\pmod {11}\\
&\equiv (1+2\cdot 5+3\cdot 6+4\cdot 9+5\cdot 4+6\cdot 7+7\cdot 3+8\cdot 0+9\cdot 3)\pmod {11}\\
&\equiv (1+10+18+36+20+42+21+27)\pmod {11}\\
&\equiv 175\pmod {11}\\
&\equiv 10\pmod {11}.\\
\end{align*}
Thus ## \sum^{9}_{k=1} ka_{k}\pmod {11}\equiv a_{10} ##.
Therefore, the ISBN ## 1-56947-303-10 ## is correct.

Math100 said:
Homework Statement:: The International Standard Book Number (ISBN) used in many libraries consists of nine digits ## a_{1}a_{2}...a_{9} ## followed by a tenth check digit ## a_{10} ##, which satisfies
## a_{10}\equiv \sum^{9}_{k=1} ka_{k}\pmod {11} ##.
Determine whether each of the ISBNs below is correct:
(a) 0-07-232569-0 (United States).
(b) 91-7643-497-5 (Sweden).
(c) 1-56947-303-10 (England).
Relevant Equations:: None.

(a)
Consider the ISBN ## 0-07-232569-0 ## from the United States.
Observe that
\begin{align*}
&\sum^{9}_{k=1} ka_{k}\pmod {11}\\
&\equiv (a_{1}+2a_{2}+\dotsb +9a_{9})\pmod {11}\\
&\equiv (0+2\cdot 0+3\cdot 7+4\cdot 2+5\cdot 3+6\cdot 2+7\cdot 5+8\cdot 6+9\cdot 9)\pmod {11}\\
&\equiv (21+8+15+12+35+48+81)\pmod {11}\\
&\equiv 220\pmod {11}\\
&\equiv 0\pmod {11}.\\
\end{align*}
Thus ## \sum^{9}_{k=1} ka_{k}\pmod {11}\equiv a_{10} ##.
Therefore, the ISBN ## 0-07-232569-0 ## is correct.

(b)
Consider the ISBN ## 91-7643-497-5 ## from Sweden.
Observe that
\begin{align*}
&\sum^{9}_{k=1} ka_{k}\pmod {11}\\
&\equiv (a_{1}+2a_{2}+\dotsb +9a_{9})\pmod {11}\\
&\equiv (9+2\cdot 1+3\cdot 7+4\cdot 6+5\cdot 4+6\cdot 3+7\cdot 4+8\cdot 9+9\cdot 7)\pmod {11}\\
&\equiv (9+2+21+24+20+18+28+72+63)\pmod {11}\\
&\equiv 257\pmod {11}\\
&\equiv 4\pmod {11}.\\
\end{align*}
Thus ## \sum^{9}_{k=1} ka_{k}\pmod {11}\not\equiv a_{10} ## because ## a_{10}=5 ##.
Therefore, the ISBN ## 91-7643-497-5 ## is not correct.

(c)
Consider the ISBN ## 1-56947-303-10 ## from England.
Observe that
\begin{align*}
&\sum^{9}_{k=1} ka_{k}\pmod {11}\\
&\equiv (a_{1}+2a_{2}+\dotsb +9a_{9})\pmod {11}\\
&\equiv (1+2\cdot 5+3\cdot 6+4\cdot 9+5\cdot 4+6\cdot 7+7\cdot 3+8\cdot 0+9\cdot 3)\pmod {11}\\
&\equiv (1+10+18+36+20+42+21+27)\pmod {11}\\
&\equiv 175\pmod {11}\\
&\equiv 10\pmod {11}.\\
\end{align*}
Thus ## \sum^{9}_{k=1} ka_{k}\pmod {11}\equiv a_{10} ##.
Therefore, the ISBN ## 1-56947-303-10 ## is correct.
I got the same result. But that only reduces the chances. We still could have made both the same error, at least modulo 11.

berkeman
fresh_42 said:
I got the same result. But that only reduces the chances. We still could have made both the same error, at least modulo 11.
What's the error then?

Math100 said:
What's the error then?
That was a joke. I don't think you made one. However, I checked the calculations only in mind.

The first book I checked here, however, has a ten-digit number plus a check digit.

Math100
Let's test ##3-411-014420-2.##

##3+4\cdot 2+ 3+4+6+4\cdot 7+4\cdot 8+2\cdot 9 \equiv 3 \not\equiv 2\pmod{11}.## Hmm ...

Math100
fresh_42 said:
Let's test ##3-411-014420-2.##

##3+4\cdot 2+ 3+4+6+4\cdot 7+4\cdot 8+2\cdot 9 \equiv 3 \not\equiv 2\pmod{11}.## Hmm ...
I know, it's a bit weird.

They say a new 13 digits code has been applied since 2007.

anuttarasammyak said:
They say a new 13 digits code has been applied since 2007.
My book example is older.

anuttarasammyak

## 1. What is an ISBN?

An ISBN (International Standard Book Number) is a unique identifier assigned to a book or other publication to help identify and track it in the publishing industry.

## 2. How many digits are in an ISBN?

An ISBN typically consists of 13 digits, but in some cases, it may have 10 digits.

## 3. How is an ISBN formatted?

An ISBN is typically divided into five parts: prefix element, registration group, registrant element, publication element, and check digit. The format for a 13-digit ISBN is: prefix element (978 or 979), registration group (0-9), registrant element (1-7 digits), publication element (1-6 digits), and check digit (1 digit). The format for a 10-digit ISBN is: prefix element (0 or 1), group identifier (1-5 digits), publisher code (1-7 digits), and title identifier (1-6 digits).

## 4. How do you determine if an ISBN is correct?

An ISBN can be checked for accuracy by using a mathematical formula called the Modulus 10 algorithm. This involves multiplying each digit of the ISBN by its position in the sequence and then taking the sum of these products. The result should be divisible by 11 for a valid ISBN.

## 5. Can an ISBN be changed or reused?

No, once an ISBN is assigned to a publication, it cannot be changed or reused. This ensures that each book has a unique identifier and can be easily tracked in the publishing industry.

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