So I heard someone say that if you have $\frac{a}{b}$, then the period will not exceed $b$ digits. He said it can be proved with pigeonhole principle, but did not provide further explanation. Can someone make me understand? Thanks.
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1Welcome to MSE! Please use the basic tutorial and quick reference guide and also show the work you have done so far. Can you also cite the textbook, or lecture notes, that you're referring to. – Jessie Feb 04 '21 at 14:35
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You probably mean, like, the period of the repeating decimal representation of $a\div b$, right? – Mike Pierce Feb 04 '21 at 14:43
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The best way to understand is to try it out by long division: first $k/7$ then $k/17$ (choose any $k\ne0$ of your liking). – Chrystomath Feb 04 '21 at 17:09
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In case you divide a number by $7$: how many different remainders are possible? Perform a long division. – Michael Hoppe Feb 04 '21 at 18:36