Modular arithmetic is often used in assigning an
**extra digit to identification numbers for the purpose of**

**detecting forgery or errors**
. For example, airline companies, UPS

and
**Avis and National rental car companies use the modulo 7**
values of identification numbers to

assign check digits, while the
**US Postal service uses modulo 9**
:

A United States Postal Service money order (mod 9) has an identification number consisting of 10 digits

together with an extra digit called a check. The check digit is the 10-digit number modulo 9.

Thus, the number 3953988164 has the check digit 2 since

`> `
**3953988164 mod 9;**

So, the number printed on the top of the money order is 39539881642. If this number were

incorrectly entered into a computer (programmed to calculate the check digit) as, say,

39559881642 (and error in the fourth position),

then the machine would calculate the check digit (the last number on the check) as

`> `
**3955988164 mod 9;**

The error would be detected if the check digit of 2 (the last number on the money order number 39559881642) does not match the computer's calculation.

Does the machine calculate the error in this case? Explain why or why not.

Suppose that a money order with the identification number and check digit 21720421168 is erroneously copied as 27750421168. Will the check digit 8 detect the error? Explain using Maple statements and a text explanation.

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