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Multiples of Four


Date: 10/22/2001 at 18:02:17
From: Jenny
Subject: Multiples of four

How many multiples of four from 1 through 1000 do not contain any of 
the digits 6, 7, 8, 9, or 0?

I don't understand this at all.


Date: 10/23/2001 at 12:55:14
From: Doctor Sarah
Subject: Re: Multiples of four

Hi Jenny - thanks for writing to Dr. Math.

We could start by listing multiples of 4 and looking for patterns. Do 
you know what a multiple of a number is?  It's that number multiplied 
by another number. So the first multiple of 4 is 4x1, the second 
multiple of 4 is 4x2=8, the third multiple of 4 is 4x3=12, etc.  

Let's make a table of multiples of 4 from 1 to 100, with columns A-E 
across the top and rows 1-5 down the left-hand side:

     A      B      C      D      E

1    4      8     12     16     20
2   24     28     32     36     40
3   44     48     52     56     60
4   64     68     72     76     80
5   84     88     92     96    100

Now let's look at these multiples, remembering that there will be nine 
more tables like this one from 101-1000.

Let's look for 6,7,8,9, and 0 in the columns first. Aha! We can erase 
all of columns B, D, and E because there's a 6 or an 8 or a 0 in each 
number in those columns.  Now we're left with just:

     A      C 

1    4     12 
2   24     32 
3   44     52 
4   64     72 
5   84     92 

Now let's look at the rows. Wow! We can eliminate rows 4 and 5 because 
they have 6, 7, 8, or 9, leaving:

     A      C 

1    4     12 
2   24     32 
3   44     52 

Just 6 numbers left!

So if from 1-100 there are 6 multiples of four that do not contain any 
of the digits 6, 7, 8, 9, or 0, how many multiples of four like this 
are there from 1-1000?

- Doctor Sarah, The Math Forum
  http://mathforum.org/dr.math/   
    
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