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Find a Four-Digit Number Divisible by 2, 3, and 9

Date: 11/29/2003 at 14:13:54
From: Bri
Subject: four digit number divisible by 2,3,9

What is a four digit number that is divisible by 2, 3, and 9?  For
example, a three digit number that is divisible by 2 and 9 is 108.

It  can't be 1,000 because 3 can't go into it.  It can't be 2,000 
because 3 and 9 can't go into it.



Date: 12/01/2003 at 14:10:02
From: Doctor Jason
Subject: Re: four digit number divisible by 2,3,&9

Hi Bri,

Thanks for writing to Dr. Math!

If a number is divisible by 9, then it is also divisible by 3, since 3
is a factor of 9 (3 * 3 = 9).  

If a number ends in 0, 2, 4, 6, or 8, then it is divisible by 2.  

So, what you need is an even, four-digit number that is divisible by
9.  (Hint: If the sum of all of the digits of a number is divisible by
9, then the number itself is divisible by 9.)

The first that comes to my mind is 9000. 

       9000 / 2 = 4500
       9000 / 3 = 3000
       9000 / 9 = 1000

To find more examples, you might just grab a calculator and enter
even, four-digit numbers and see if they are divisible by 9.  

You could also just use all of the rules that we have listed above to
"create" a number that fits the 3 criteria.

1.  Start with four blank spaces representing the 4 digits in the number: 
              ___  ___  ___  ___

2.  Fill in the ones digit with an even number, which takes care of
the 'must be divisible by 2' rule:
   
              ___  ___  ___  _4_ 

3.  Next, fill in 2 of the remaining blanks with any numbers you want:  

              _2_  _7_  ___  _4_

4.  Now add the 3 digits together.  Subtract your sum from either 9,
18, 27, whichever is the next multiple of 9 that is greater than your
sum.  The difference is the number that will fill the last blank.  So
far, I have used 2, 7, and 4.  The sum of those three digits is 13.  I
need to subtract 13 from 18, since 18 is the next multiple of 9 that
is greater than 13: 18 - 13 = 5.  The number 5 fills my last blank:

              _2_  _7_  _5_  _4_

5.  Check your answer: 

     2754 / 2 = 1377
     2754 / 3 = 918
     2754 / 9 = 306

Note that as long as we keep an even number in the ones position, we
can rearrange the digits for even more possibilities since the sum of
the digits will still be divisible by 9:

     2574 / 9 = 286
     5274 / 9 = 586
     5724 / 9 = 636
     4572 / 9 = 508
     4752 / 9 = 528 
     7542 / 9 = 838
     7524 / 9 = 836
     7254 / 9 = 806
     7425 / 9 = 825

Try it for yourself and let me know how it works out for you!
              
- Doctor Jason, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Elementary Division
Middle School Division

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