1-3-20-have-how-many-digits-in-periodic-part-

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Question Number 400 by 123456 last updated on 25/Jan/15

$$\frac{1}{3^{20}}\mathrm{have}\mathrm{how}\mathrm{many}\mathrm{digits}\mathrm{in}\mathrm{periodic}\mathrm{part}?$$

Answered by prakash jain last updated on 29/Dec/14

$$3^{20}\mathrm{is}\mathrm{not}\mathrm{divisible}\mathrm{by}2\mathrm{or}5.\mathrm{So}$$$$\mathrm{periodicity}\mathrm{is}\mathrm{given}\mathrm{by}\mathrm{equation}$$$${10}^n\equiv 1\left(\mathrm{mod}{3}^{20}\right)$$$$n\mathrm{is}\mathrm{multiplicative}\mathrm{order}\mathrm{and}$$$$\mathrm{it}\mathrm{given}\mathrm{the}\mathrm{periodicity}\mathrm{of}\frac{1}{3^{20}}$$$$\mathrm{Algorithmic}\mathrm{calculation}\mathrm{of}n\mathrm{given}$$$$n=387420489$$$$\mathrm{The}\mathrm{number}\mathrm{of}\mathrm{digits}\mathrm{in}\mathrm{periodic}$$$$\mathrm{part}=387420489$$

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