Casting Out Nines

By Unknown

Monday, October 11, 2010

Did you know that the digits of any multiple of 9 add up to a multiple of 9?* For example, 9 * 125 = 1125, and 1+1+2+5 = 9.  And that's the basis for lots of number tricks.

Here's a version of an online trick using a crystal ball to predict your number.

But some of the tricksters aren't careful enough about their math, and can mess up. Shecky, at Math Frolic, pointed to this one (get your calculator out), that tells you your original number, and the digits you chose later in the process. But it messed up for me. I think it will mess up about one-tenth of the time, actually.

Here's how it goes:

On a PIECE of PAPER, write a number between 10 and 10,000.

I chose 13.

Multiply it by 4.

52

Now add 5 to the result.

57

Now multiply the result by 75.

4275

Now choose any TWO digits from the result and add them to the result.
Example: 591673 + 69

I chose the 5 and 2, and added 52.

Now multiply the result by 3.
Lets multiply the result by 3 again to make it more difficult.

I multiplied by 9 to get 38943.

Now substitute one digit with an "X" and enter your result below.

I chose the 9.

The result was this:

 

The first number was: 12
Then you choose: 52
And you substituted the number 0 with an "X"

 

If I went back one step and put the x in for the 8 instead, it got everything right. 

 

What went wrong? 

 

By the way, casting out nines refers to a way to check your addition (important back when we didn't use calculators and computers). It depends on the fact that the remainder after division by nine is always the same for a number and the sum of its digits. 

______
*Why?

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