Even or Odd in Base 5?

Date: 09/23/1999 at 14:04:23
From: Susan
Subject: Different Number Bases

Could you describe a rule for determining whether a number written in 
the base five number system is even or odd, without converting it to 
base ten?

Date: 09/23/1999 at 14:53:04
From: Doctor Mitteldorf
Subject: Re: Different Number Bases

Do you know how casting out nines works in base 10? You can add up the 
digits of any number, and the number is divisible by nine if and only 
if the sum of the digits is divisible by nine.

The same kind of thing works with base 5, where you "cast out fours." 
The number is divisible by 4 if and only if the sum of the digits is 
divisible by 4.

You can make things a little easier yet and derive with very similar 
reasoning a rule that says: If a number in base 5 has an even number 
of odd digits, then it is divisible by 2, but if it has an odd number 
of odd digits, it is not.

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