All books are assigned a 10 digit ISBN# (d_{10}d_9d_8...d_2d_1) which has the following property: sum_{i=1}^{10}id equiv 0(mod (11))

Talon Mcbride 2022-07-17 Answered
All books are assigned a 10 digit ISBN ( d 10 d 9 d 8 . . . d 2 d 1 ) which has the following property:
i = 1 10 i d i 0 ( mod ( 11 ) )
Prove that if you swap two adjacnt digits in an ISBN#, it is no longer a valid ISBN#.
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Answers (2)

Damarion Pierce
Answered 2022-07-18 Author has 11 answers
Step 1
Let S 1 be the "right" sum and S 2 be the sum with d i and d i + 1 transposed.
Then S 1 S 2 = i d i + ( i + 1 ) d i + 1 ( i d i + 1 + ( i + 1 ) d i ) = d i + 1 d i
Step 2
Since S 1 0 ( mod 11 ) you have that S 2 d i + 1 d i ( mod 11 ), or better S 2 d i d i + 1 ( mod 11 )
If d i d i + 1 ( mod 11 ) then the S 2 is obviously not the sum of a valid ISBN since S 2 0 ( mod 11 ).
On the other hand, if they are the same digit mod 11, then transposing them makes no difference, and you do get a valid word. This is a inaccuracy in the problem statement. For example, the all zeros ISBN number is still valid no matter how many transpositions you do.
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Paxton Hoffman
Answered 2022-07-19 Author has 6 answers
Step 1
Suppose you swap d n and d n + 1
Then the sum becomes ( i = 1 10 d i ) + d n d n + 1
Since i = 1 10 d i 0 ( mod 11 )
Step 2
For the new sum to be divisible by 11,
d n d n + 1 0 ( mod 11 )
d n d n + 1 k ( mod 11 ) where 0 k 9 and to be divisble by 11 the 2 adjacent digits must be same which means that swapping, in this case, doesn't change the ISBN.
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