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### 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
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Permutations and Combinations

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