<Li> UPC - A can detect about 89% of transposition errors . Specifically, if and only if the difference between two adjacent digits is 5, the UPC - A can't detect their transposition . <Ol> <Li> If 2 neighboring digits are transposed, then one of the digits a will be weighted by 1, and the other digit b = a + d will be weighted by 3, where d is the difference between the two digits . If the digits were in their correct order, they would contribute <Dl> <Dd> <Dl> <Dd> 1 a + 3 b = 1 a + 3 (a + d) = 4 a + 3 d (\ displaystyle 1a + 3b = 1a + 3 (a + d) = 4a + 3d) </Dd> </Dl> </Dd> <Dd> to the left hand side of the check digit equation . In the transposed order, they contribute <Dl> <Dd> 1 b + 3 a = 3 a + 1 (a + d) = 4 a + d (\ displaystyle 1b + 3a = 3a + 1 (a + d) = 4a + d). </Dd> </Dl> </Dd> <Dd> to the LHS . Subtracting the two contributions gives how much they change the LHS: <Dl> <Dd> (4 a + 3 d) − (4 a + d) = 2 d (\ displaystyle (4a + 3d) - (4a + d) = 2d) </Dd> </Dl> </Dd> <Dd> An error will be detected as long as the modular change is nonzero; if 2d ≡ 0 modulo 10, then the change will not be detected . Consequently, only when the character difference d ≡ 5 will an error be undetected (when d ≡ 0 the degenerate "transposition" is not an error). </Dd> </Dl> </Li> <Li> Next consider how often a transposition has a distance d of 5 . </Li> </Ol> </Li> <Ol> <Li> If 2 neighboring digits are transposed, then one of the digits a will be weighted by 1, and the other digit b = a + d will be weighted by 3, where d is the difference between the two digits . If the digits were in their correct order, they would contribute <Dl> <Dd> <Dl> <Dd> 1 a + 3 b = 1 a + 3 (a + d) = 4 a + 3 d (\ displaystyle 1a + 3b = 1a + 3 (a + d) = 4a + 3d) </Dd> </Dl> </Dd> <Dd> to the left hand side of the check digit equation . In the transposed order, they contribute <Dl> <Dd> 1 b + 3 a = 3 a + 1 (a + d) = 4 a + d (\ displaystyle 1b + 3a = 3a + 1 (a + d) = 4a + d). </Dd> </Dl> </Dd> <Dd> to the LHS . Subtracting the two contributions gives how much they change the LHS: <Dl> <Dd> (4 a + 3 d) − (4 a + d) = 2 d (\ displaystyle (4a + 3d) - (4a + d) = 2d) </Dd> </Dl> </Dd> <Dd> An error will be detected as long as the modular change is nonzero; if 2d ≡ 0 modulo 10, then the change will not be detected . Consequently, only when the character difference d ≡ 5 will an error be undetected (when d ≡ 0 the degenerate "transposition" is not an error). </Dd> </Dl> </Li> <Li> Next consider how often a transposition has a distance d of 5 . </Li> </Ol> <Li> If 2 neighboring digits are transposed, then one of the digits a will be weighted by 1, and the other digit b = a + d will be weighted by 3, where d is the difference between the two digits . If the digits were in their correct order, they would contribute <Dl> <Dd> <Dl> <Dd> 1 a + 3 b = 1 a + 3 (a + d) = 4 a + 3 d (\ displaystyle 1a + 3b = 1a + 3 (a + d) = 4a + 3d) </Dd> </Dl> </Dd> <Dd> to the left hand side of the check digit equation . In the transposed order, they contribute <Dl> <Dd> 1 b + 3 a = 3 a + 1 (a + d) = 4 a + d (\ displaystyle 1b + 3a = 3a + 1 (a + d) = 4a + d). </Dd> </Dl> </Dd> <Dd> to the LHS . Subtracting the two contributions gives how much they change the LHS: <Dl> <Dd> (4 a + 3 d) − (4 a + d) = 2 d (\ displaystyle (4a + 3d) - (4a + d) = 2d) </Dd> </Dl> </Dd> <Dd> An error will be detected as long as the modular change is nonzero; if 2d ≡ 0 modulo 10, then the change will not be detected . Consequently, only when the character difference d ≡ 5 will an error be undetected (when d ≡ 0 the degenerate "transposition" is not an error). </Dd> </Dl> </Li> <Dl> <Dd> <Dl> <Dd> 1 a + 3 b = 1 a + 3 (a + d) = 4 a + 3 d (\ displaystyle 1a + 3b = 1a + 3 (a + d) = 4a + 3d) </Dd> </Dl> </Dd> <Dd> to the left hand side of the check digit equation . In the transposed order, they contribute <Dl> <Dd> 1 b + 3 a = 3 a + 1 (a + d) = 4 a + d (\ displaystyle 1b + 3a = 3a + 1 (a + d) = 4a + d). </Dd> </Dl> </Dd> <Dd> to the LHS . Subtracting the two contributions gives how much they change the LHS: <Dl> <Dd> (4 a + 3 d) − (4 a + d) = 2 d (\ displaystyle (4a + 3d) - (4a + d) = 2d) </Dd> </Dl> </Dd> <Dd> An error will be detected as long as the modular change is nonzero; if 2d ≡ 0 modulo 10, then the change will not be detected . Consequently, only when the character difference d ≡ 5 will an error be undetected (when d ≡ 0 the degenerate "transposition" is not an error). </Dd> </Dl>

Where do i find the upc on a product