Divisibility by Four

Date: 04/19/97 at 19:55:32
From: Marianne Unruh
Subject: Number

How many 6-digit numbers are divisible by 4 if we allow no repeated 
digits?

Date: 04/20/97 at 15:26:37
From: Doctor Steven
Subject: Re: Number

We have a 6-digit number with no digit repeated and we want to know if 
it's divisible by 4.

We are only worried about the last two digits since the first four 
digits are multiplied by 100, which is divisible by 4.

                       ?   ?
      --- --- --- --- --- ---

Well, we know that the last digit has to be even since no odd number 
is divisible by 4.  So for the last digit we have 5 choices: 0, 2, 4, 
6, or 8.

If we pick a number that is divisible by 4, then we need another even 
number for the second digit. So we would have 3 choices for the ones 
digit (since we can't have two of the same number) and then 4 choices 
for the tens digit. Then we would pick any sequence of digits for the 
hundreds and higher digits. This gives us 12*1680 = 20160 numbers 
divisible by 4.
                   
If we pick an even number that is not divisible by 4, then we need an 
odd number for the tens place. So we have 2 choices for our ones 
digits and 5 choices for our tens digit. Then we would again pick any 
sequence to fill in the remaining four digits. So we would get 
10*1680 = 16800 numbers divisible by 4 this way.

Summing these numbers up we get that there are 36,960 6-digit numbers 
divisible by 4 if we don't use any number twice.

-Doctor Steven,  The Math Forum
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