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### When Casting Out Nines Fails

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Date: 11/29/2001 at 21:05:19
From: Teresa Miller
Subject: Casting out 9's, division

My son is doing Abeka 6th grade math. I was intrigued with checking
math by casting out 9's. I introduced this technique to a friend.
Just to prove that casting out 9's worked, I intentionally used an
incorrect answer (quotient). I was shocked when the problem checked
out correctly. Here is the problem. Divisor 6, Dividend 5223, and
Quotient 875. Have I done something wrong? I know the correct answer
is 870 R. 3.

Really stumped,
Teresa Miller
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Date: 11/29/2001 at 23:30:30
From: Doctor Peterson
Subject: Re: Casting out 9's, division

Hi, Theresa.

It's not at all surprising that casting out nines would fail; it puts
all numbers into one of 9 categories, so it will catch an error that
puts it into one of the 8 wrong categories, but not one that happens
to land it back in the "right" category. This means there is a 1 in 9
chance that a random error will look okay. When the method is taught,
it should always be pointed out that it can tell you if an answer is
wrong, but can't be trusted to tell you that it is right!

Claim: 875 * 6 + 0 = 5223
Check:  8  * 6 + 0 =? 3
48
3 =? 3

(There are different ways to arrange the work, of course. I calculated
the digit sums, then calculated the left side as 48 and found its
digit sum to be 3.)

The correct answer, checked, would be

Claim: 870 * 6 + 3 = 5223
Check:  6  * 6 + 3 =? 3
39
3 =? 3

You have added 5 to the quotient, which adds 5*6 (or 3) to the digit
sum; and you dropped the remainder of 3, which subtracted 3 from the
digit sum, so you came out even.

In case you are interested, here is my explanation of the method:

Casting Out Nines to Check Arithmetic
http://mathforum.org/library/drmath/view/55926.html

I checked, and I do explain the pitfalls of the method. Whew!

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