Three-Digit Numbers

Date: 01/27/97 at 18:56:34
From: Justine
Subject: Three-digit numbers

If I take a three-digit number and rearrange it two other ways, add 
the three numbers together, and divide by 3, I always get a remainder 
of zero.  Why is that?

Date: 01/27/97 at 20:05:05
From: Doctor Wilkinson
Subject: Re: Three-digit numbers

Very good question.  This depends on some facts about how remainders 
work.  First of all, if you take two numbers, add them, and divide by 
3, you always get the same remainder as if you had divided each number 
by three, taken the remainders, added them together, divided the sum 
by three and taken the remainder.  

For example, take 5 and 8.

   Add and take the remainder:  13 divided by 3, remainder is 1.

   Remainder from dividing 5 by 3 is 2
   Remainder from dividing 8 by 3 is 2
   Add and take the remainder:  4 divided by 3, remainder is 1.

This part works for any number, not just 3.

Second, the numbers 10, 100, 1000, 10000, and so on, all leave a 
remainder of 1 when you divide them by 3.

Now take any 3-digit number, like 157.  Rearrange it two ways, say 175 
and 751.  Now when you add these three numbers together, you're going 
to get the same remainder when you divide by 3 as if you just added 
the digits of the numbers.  That is, instead of adding 157, 175, 751 
you can just add

 1 + 5 + 7 + 1 + 7 + 5 + 7 + 5 + 1

This is because of the fact that 50, for example, leaves the same 
remainder as 5, and 700 leaves the same remainder as 7.

But now you have three 1's and three 7's and three 5's.  When you add 
three of the same thing, the result is going to leave a reaminder of 0 
when you divide by 3, so the whole sum is going to leave a remainder 
of 0 when you divide by 3.

Keep up the good work.  Keep asking questions.

-Doctor Wilkinson,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/