On 10/25/09, Rich/CA/Math wrote:

This reminds me of a puzzle a student's dad gave her last year, and that she
asked me for help on. I'm drawing a blank now but I'll have to find it and
then I can use this as an enrichment when we start talking about rational
and irrational numbers. Coolo.
>
> A fraction is a division problem. When you do a long division problem, in
> each cycle there is either a remainder or not. If there is no remainder
> then the decimal terminates.
>
> But lets say there is never no remainder, i.e., the decimal never
terminates.
>
> If the denominator is n then there are at most n-1 possible remainders.
> For example, if the denominator is 7 then the only possible remainders are
> 1, 2, 3, 4, 5, or 6. Once a remainder repeats a previous remainder the
> decimal will repeat from that point forward.
>
> So at the latest, when you get to the 7th cycle of the long division, you
> will have to have a repeat of the remainder. And the decimal repeats.
>
> This also proves a corollary, that the length of the repeating portion
> cannot exceed the denominator minus 1.
>
> Try it with 1/7. In the first cycle the division result is 1 and the
> remainder is 3. In the next cycle the division result is 4 and the
> remainder is 2. In the third cycle the division result is 2 and the
> remainder is 6. In the fourth cycle the division result is 8 and the
> remainder is 4. In the fifth cycle the division result is 5 and the
> remainder is 5, In the sixth cycle the division result is 7 and the
> remainder is 1. And that essentially repeats the one that you were
> originally dividing into. So the pattern of remainders will repeat,
> 3,2,6,4,5,1 and as they do the division results: 1, 4, 2, 8, 5, 7 will
> repeat as well.
>
> I think this will be accessible to your brighter students.
>
> This can actually also be used as on introduction to the pigeon-hole
principle.

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