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### Accountants Use 9 to Check for Errors

```
Date: 02/21/2002 at 23:11:51
From: Steven Etheridge
Subject: Number theory

I know accountants divide the difference of debits and credits by 9
to check for a transposition error. I need to understand why this
works. It may be related to casting out nines, etc., but I can't quite
figure it out.

Thanks!
```

```
Date: 02/22/2002 at 12:31:20
From: Doctor Schwa
Subject: Re: Number theory

Hi Steven,

The material on Divisibility on our frequently asked questions page,

http://mathforum.org/dr.math/faq/

a number is divisible by 9.

The idea is that if you have digits a,b in some number, with place
value the "a" is "worth" a * 10^n and b is worth b * 10^(n-1). So in
total the ab is worth a * 10^n + b * 10^(n-1), or 10^(n-1) * (10a + b)

When you switch them, the "ba" is then worth 10^(n-1) * (10b+a).

So the difference is 10^(n-1) * (9a - 9b), which is sure to be
divisible by 9.

The most common other type of error is a single-digit change, which
will only be divisible by 9 if you switch a 9 with a 0, so most of the
time, if your error is divisible by 9, it's due to a transposition.

- Doctor Schwa, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory

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