Even and Odd Numbers in Base 5

Date: 02/02/2002 at 15:35:59
From: Sheri Griffiths
Subject: Even and odd numbers in base 5

How can you tell if a number in base 5 is even or odd?

I know that all even numbers in base 10 are divisible by 2, but I 
don't know if it translates over exactly to base 5.  I want to know 
if there are conversions from base 10 to base 5 to be able to figure 
out this answer.

Date: 02/02/2002 at 17:32:32
From: Doctor Rick
Subject: Re: Even and odd numbers in base 5

Hi, Sheri.

I think of "even" as meaning "divisible by 2," and this is independent 
of the base in which we represent a number. But perhaps you are 
thinking of "even" as meaning "having a units digit that is divisible 
by 2." It's true that not all numbers whose units digit in base 5 is 
even are themselves even (that is, divisible by 2). The even numbers 
are

  Base 10: 0, 2, 4,  6,  8, 10, 12, 14, 16, 18, ...
  Base 5:  0, 2, 4, 11, 13, 20, 22, 24, 31, 33, ...

You will notice that in every case in this list, the SUM of the digits 
of the base-5 numeral is even. We can't say yet that this rule holds 
true, because we haven't looked at any 3-digit numerals; and even then 
we won't be sure until we actually prove the rule.

Here is how we can prove our rule. The numeral abcd in base 5 
represents the number 

  125a + 25b + 5c + d

The powers of 5 are all odd, so we can write the number like this:

  (2*62+1)a + (2*12+1)b + (2*2+1)c + d

  2(62a+12b+2c+d) + (a+b+c+d)

The first term must be even. If the second term is even, then abcd 
will be even; and if the second term is odd, then abcd will be odd. 
Thus our hypothesized rule works: a numeral in base 5 is even or odd 
according as the sum of its digits is even or odd. Since *all* powers 
of 5 are odd, the proof can be extended to any number of digits. In 
fact, the same rule holds for all odd bases. Do you see why?

We have a briefer answer to this question, with a nice addition that 
makes the rule easier to use, in our Dr. Math Archives:

   Even or Odd in Base 5?
   http://mathforum.org/dr.math/problems/susan.9.23.99.html   

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