Divisibility and Subtraction in Other Bases

Date: 07/28/99 at 00:37:11
From: Agnes Morrison
Subject: Number bases other than 10

In base ten you can tell when a number is even by looking at its last 
digit. Can you recognize an even number in other bases, for example 
base 2 or 3 or 4? 

Thank you for the help.

Date: 07/28/99 at 11:34:10
From: Doctor Rick
Subject: Re: Number bases other than 10

Hi, Agnes. This question is a nice topic for exploration!

Look for yourself. I'll write out some numbers in base 10, 2, 3, 
and 4:

     base 10  base 2  base 3  base 4
     -------  ------  ------  ------
        1        1       1       1
        2       10       2       2
        3       11      10       3
        4      100      11      10
        5      101      12      11
        6      110      20      12
        7      111      21      13
        8     1000      22      20
        9     1001     100      21

In base 2, all even numbers end in 0 (the only even digit) and odd 
numbers end in 1 (the only odd digit), so the rule works in base 2. 
What about bases 3 and 4 - which bases work this way? Can you come up 
with a hypothesis (an idea that you can test) about which bases the 
rule works for?

Let's not stop here. Evenness is the same as divisibility by 2. What 
about divisibility by other numbers?

In base 10, you can test divisibility by 5 in the same way as 
divisibility by 2: if the last digit is divisible by 5 then the whole 
number is divisible by 5. Do you notice something here? The base, 10, 
is divisible by 2 and by 5. Those are the two numbers whose 
divisibility you can test by looking at the last digit of a number in 
base 10.

Can you guess what numbers you can do this with in base 3? Base 8? 
Base 12? Test your hypotheses by writing numbers in these bases as I 
did. Then try to understand why it works. I'd be glad to correspond 
with you further if you have ideas.

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

Date: 07/28/99 at 16:31:55
From: Marc Morrison
Subject: Re: Number bases other than 10

Dear Dr. Rick,

Thank you for the neat answer. Can the idea be that if the base can be 
divided evenly by two, then you can tell by looking to see if the last 
digits are even numbers that the number is an even number? And too, if 
the base is evenly divisible by 2, and the last digit is an odd 
number, then the number is odd? If the base number is not evenly 
divisible by 2, then you can't tell by looking at the last digit 
whether the number is even or odd. Does this work?

Thank you again for your help.
Agnes

Date: 07/28/99 at 18:04:48
From: Doctor Rick
Subject: Re: Number bases other than 10

Hi, Agnes -

Yes, you've got the idea. What do you think about divisibility by 3? 
And why do you think this works? (Hint: 5274 = 527 * 10 + 4; have you 
learned about the distributive property yet?)

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

Date: 07/28/99 at 18:11:00
From: Marc Morrison
Subject: Re: Number bases other than 10

Dear Dr. Rick,

The last part of my project is to learn subtraction in other bases. I 
get confused. What I am supposed to borrow? Can you help some more?

Thank you.
Agnes

Date: 07/29/99 at 11:04:49
From: Doctor Rick
Subject: Re: Number bases other than 10

Hi again, Agnes.

Subtraction in other bases isn't very hard; I'll just give you some 
examples. The basic rule is that you always borrow the base. For 
instance, in base 4:

       312
     - 133
     -----

You can't subtract 3 from 2, so you need to borrow. Take 1 from the 
4's column and change it to four 1's; add the four 1's to the two 1's 
that you have:

       3 0 6
     - 1 3 3
     -------
           3

Now we move to the 4's column. We can't subtract 3 from 0, so we need 
to borrow again. Take 1 from the 16's column and change it to four 
4's; add the four 4's to the zero 4's that you have:

       2 4 6
     - 1 3 3
     -------
         1 3

Finally we move to the 16's column, where we have no problem 
subtracting 1 from 2:

       2 4 6
     - 1 3 3
     -------
       1 1 3

Let's just check our answer by adding 113 to 133 in base 4. You can 
check your understanding of adding in base 4:

       133
     + 113
     -----
         6

6 doesn't belong in base 4; we need to break it up into 4 + 2, and 
carry the four 1's as one 4.

        1
       133
     + 113
     -----
         2

1 + 3 + 1 = 5, which again doesn't belong in base 4. Break it into 
4 + 1 and carry the four 4's as one 16:

       11
       133
     + 113
     -----
        12

Finally we add 1 + 1 + 1 = 3, and no carry is needed:

       11
       133
     + 113
     -----
       312

That's the correct answer: we subtracted 133 from 312 to get 113, then 
we added 133 back on to 113 and got 312 back - all in base 4.

Does this make sense?

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

Date: 07/29/99 at 11:30:36
From: Marc Morrison
Subject: Re: Number bases other than 10

Dear Doctor Rick,
Thank you a bunch for all your help. You are great!

Agnes