1. ## Barcode proof problem

A particular identification code (ID) has a 9-digit code with check vector c = [9, 8, 7, 6, 5, 4, 3, 2, 1]. A valid ID v has c · v = 0 in Z11.
A common error when entering an ID on machine-readable forms is to enter 0
instead of 9, or x + 1 instead of x for 0 ≤ x ≤ 8. (So a 0 gets entered as 1, or 1 as 2, for example.)
Prove that a single error of this type will always be detected. (That is, prove that if a single error of this type is made, and the resulting incorrect ID is v, then c · v is not equals to 0 in Z11.)

2. Originally Posted by platon
A particular identification code (ID) has a 9-digit code with check vector c = [9, 8, 7, 6, 5, 4, 3, 2, 1]. A valid ID v has c · v = 0 in Z11.
A common error when entering an ID on machine-readable forms is to enter 0
instead of 9, or x + 1 instead of x for 0 ≤ x ≤ 8. (So a 0 gets entered as 1, or 1 as 2, for example.)
Prove that a single error of this type will always be detected. (That is, prove that if a single error of this type is made, and the resulting incorrect ID is v, then c · v is not equals to 0 in Z11.)