12345679 Puzzle - The Missing Eight

Date: 08/17/2001 at 20:09:32
From: Michelle
Subject: The Missing Eight

I can't figure out why this works. Here it is:

You tell people to write down the numbers 12345679 (with no eight). 
Then you tell them to cross out the number they hate the most. 

You tell them that if they chose 1, to multiply 12345679 by 9; 
if it's 2, multiply by 18; if 3, by 27; if 4, by 36; and so on 
(multiples of 9).

When they're done (say they crossed out 3), they get 333333333.

How do you get that answer? How does it work?

Date: 08/17/2001 at 22:30:05
From: Doctor Peterson
Subject: Re: The Missing Eight

Hi, Michelle.

What you're saying is that if I choose any number from 1 to 9, and 
multiply 12345679 by 9 times that, I will get 111111111 times my 
number. In other words,

     12345679 * 9 * n = 111111111 * n

Try multiplying 12345679 by 9, and you will get 111111111. Knowing 
this, the trick doesn't look very mysterious, does it?

What's happening is simply the associative property of multiplication. 
We're first multiplying 9 times our number, and then multiplying by 
12345679, so it's not obvious that we are really just multiplying our 
number by 111111111.

- Doctor Peterson, for the Math Forum

Date: 08/15/2001 at 21:04:43
From: Nitu Begur
Subject: This is a Math Trick

Dear Dr. Math,

This is my problem:

Start with 12345679.

Pick a number that you don't like (for example, pick 9).
Multiply that number by 9 (equals 81).
Then multiply 81 by 12345679 and the answer is: 999999999

I don't understand how you get that - I don't understand what the 
trick is.

Date: 08/16/2001 at 12:22:41
From: Doctor Ian
Subject: Re: This is a Math Trick

Hi Nitu,

If I understand you correctly, if we choose a digit n, and multiply by 
9, and then by 12345679, we end up with 'nnnnnnnnn'.  

Looking at it another way, this means that 

  nnnnnnnnn / 12345679 = 9 * n 

Do you see why this is true? 

Now, 

  nnnnnnnnn = n * 111111111

so

  n * 111111111 / 12345679 = 9 * n

which means that 

      111111111 / 12345679 = 9

which happens to be true.  That is,

  12345679 = 111111111/9

So here's how the trick works. Pick a digit, n. Multiply it by 9. Then
multiply it by 111111111/9. What do you get?

  n * 9 * (111111111/9) = n * 111111111

because the 9's cancel each other. 

Does this help?  Write back if you'd like to talk about this some 
more, or if you have any other questions. 
  
- Doctor Ian, for the Math Forum