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Question Number 400 by 123456 last updated on 25/Jan/15
(1/3^(20) ) have how many digits in periodic part?
$$\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{20}} }\:\mathrm{have}\:\mathrm{how}\:\mathrm{many}\:\mathrm{digits}\:\mathrm{in}\:\mathrm{periodic}\:\mathrm{part}? \\ $$
Answered by prakash jain last updated on 29/Dec/14
3^(20) is not divisible by 2 or 5. So periodicity is given by equation 10^n ≡1(mod 3^(20) ) n is multiplicative order and it given the periodicity of (1/3^(20) ) Algorithmic calculation of n given n=387420489 The number of digits in periodic part=387420489
$$\mathrm{3}^{\mathrm{20}} \:\mathrm{is}\:\mathrm{not}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{2}\:\mathrm{or}\:\mathrm{5}.\:\mathrm{So} \\ $$$$\mathrm{periodicity}\:\mathrm{is}\:\mathrm{given}\:\mathrm{by}\:\mathrm{equation} \\ $$$$\mathrm{10}^{{n}} \equiv\mathrm{1}\left(\mathrm{mod}\:\mathrm{3}^{\mathrm{20}} \right) \\ $$$${n}\:\mathrm{is}\:\mathrm{multiplicative}\:\mathrm{order}\:\mathrm{and}\: \\ $$$$\mathrm{it}\:\mathrm{given}\:\mathrm{the}\:\mathrm{periodicity}\:\mathrm{of}\:\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{20}} } \\ $$$$\mathrm{Algorithmic}\:\mathrm{calculation}\:\mathrm{of}\:{n}\:\mathrm{given} \\ $$$${n}=\mathrm{387420489} \\ $$$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{digits}\:\mathrm{in}\:\mathrm{periodic} \\ $$$$\mathrm{part}=\mathrm{387420489} \\ $$

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