### 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:

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.