Copyright © University of Cambridge. All rights reserved.

## 'Check Code Sensitivity' printed from http://nrich.maths.org/

Check codes are designed to pick up common errors such as
transposing two digits or miscopying a single digit. A person types
the code number into a machine which decides whether it is a valid
code or not. If someone types in a US Postal Service Money Order
number and makes a single error, just one mistake in one digit,
will the machine pick up every error of this type? Will a machine
always pick up an error in a single digit for an airline ticket
number?

US Postal
Service Money Order: This is an eleven digit number using
digits 1,2,...9 where the sum of the first ten digits is congruent
to the eleventh digit modulo 9. That is $a_1a_2\cdots a_{11}$ where
$a_1+ \cdots +a_{10} \equiv a_{11}$ mod $9$.

Airline
tickets: This number can be any length. It uses the digits 0
to 9 and the last digit is a check digit. The number formed by
omitting the check digit must be congruent to the check digit
modulo 7.

That is $a_1\cdots a_na_{n+1}$ where $a_1a_2\cdots a_n \equiv
a_{n+1}$ mod 7.