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Question Number 19002 by tawa tawa last updated on 02/Aug/17
what is the maximum number of  time three divides 333^(505)
$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{number}\:\mathrm{of}\:\:\mathrm{time}\:\mathrm{three}\:\mathrm{divides}\:\mathrm{333}^{\mathrm{505}} \\ $$
Answered by RasheedSindhi last updated on 03/Aug/17
333=37×3^2   333^(505) =37^(505) ×(3^2 )^(505) =37^(505) ×3^(1010)   3^(1010) ∣ 333^(505)   1010 times.
$$\mathrm{333}=\mathrm{37}×\mathrm{3}^{\mathrm{2}} \\ $$$$\mathrm{333}^{\mathrm{505}} =\mathrm{37}^{\mathrm{505}} ×\left(\mathrm{3}^{\mathrm{2}} \right)^{\mathrm{505}} =\mathrm{37}^{\mathrm{505}} ×\mathrm{3}^{\mathrm{1010}} \\ $$$$\mathrm{3}^{\mathrm{1010}} \mid\:\mathrm{333}^{\mathrm{505}} \\ $$$$\mathrm{1010}\:\mathrm{times}. \\ $$
Commented by tawa tawa last updated on 03/Aug/17
God bless you sir.
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$

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