Determine whether each of the ISBNs below is correct

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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.
 
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  • #2
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. :cool:
 
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  • #3
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. :cool:
What's the error then?
 
  • #4
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.
 
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  • #5
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 ...
 
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  • #6
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.
 
  • #7
They say a new 13 digits code has been applied since 2007.
 
  • #8
anuttarasammyak said:
They say a new 13 digits code has been applied since 2007.
My book example is older.
 
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