18 October, 2009

Odds & Evens...and Five.

Following on, then, from thinking about treven, trod, and trud numbers, i have been thinking about odd and even ones. And realised something a bit, well, odd.

An even number, of course, is a whole number that is divisible by two exactly, with no remainder. Obviously, an odd one is one of the rest, leaving a remainder of one. It is easy to tell an odd number from an even one, as they always end in 1, 3, 5, 7, or 9; in addition, any number which ends with one of those digits is also odd, which is not quite saying the same thing. The reverse is, of course, also true: All even numbers end in 0, 2, 4, 6, or 8.

Except...except this is only true for as long as we use base ten. Let's switch for a second to base five, that is we can use only the first five digits (counting from zero) to represent numbers. We count this way in base five: 0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20 (that's zero to ten). In this system, while the same definition of odd and even apply (divisibility by two), the representation is different. The even numbers are 0, 2, 4, 11, 13, 20 (that's zero, two, four, six, eight, ten). Evidently, in base five, the sum of the digits can be used to tell if a number is odd or even. Or, to be more precise, because it's all i've really shown here, the sum of the last two digits of a base five number shows if the number is even.

What about treven numbers, then? Is there an easy way in base five representation to tell if a number is treven, trod or trud? Well, the three times table, base five, is 3, 11, 14, 22, 30, 33, 41, 44.... I may be mistaken, but i don't see an obvious method. Why not? What is it about base ten which makes it work for three, but not base five? What numbers might it work for, base five? And what about other bases? Oh, so many questions, and all of them beyond me.

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