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Check Codes

Details are given of how check codes are constructed (using modulus arithmetic for passports, bank accounts, credit cards, ISBN book numbers, and so on. A list of codes is given and you have to check if they are valid identification numbers?

Check Code Sensitivity

You are given the method used for assigning certain check codes and you have to find out if an error in a single digit can be identified.

Transposition Fix

Age 14 to 16 Challenge Level:

An ISBN Number

Suppose an operator types a US Bank check code into a machine and transposes two adjacent digits (i.e. swaps the order of two adjacent digits) for example the operator types $a_3a_2$ instead of $a_2a_3$, will the machine pick up every error of this type? Does the same apply to ISBN numbers; will a machine detect transposition errors in these numbers?

US Bank check codes are nine digit identification numbers $a_1a_2\cdots a_{9}$ using the digits 0 to 9 where
$7a_1 + 3a_2 + 9a_3 + 7a_4 + 3a_5 + 9a_6 + 7a_7 + 3a_8 \equiv a_9$ mod 10.

ISBN Numbers have ten digits $a_1\cdots a_{10}$ using the digits 0 to 9 where
$10a_1+9a_2+8a_3+\cdots +3a_8+2a_9+a_{10} \equiv 0$ mod 11.