let-d-n-denote-the-sum-of-the-digits-of-n-i-e-d-1000-1-d-999-27-find-minimum-k-such-that-d-d-k-times-d-5-10-100-10-

Tinku Tara June 4, 2023 Arithmetic 0 Comments

Question Number 153657 by talminator2856791 last updated on 09/Sep/21

let d(n) denote the sum of the digits of n. i.e d(1000) = 1 , d(999) = 27 find minimum k such that d(d(....._(k times) d(5^(10^(100) ) ).....) ≪ 10

$$\: \\ $$$$\:\mathrm{let}\:{d}\left({n}\right)\:\mathrm{denote}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\: \\ $$$$\:\mathrm{of}\:{n}. \\ $$$$\:\mathrm{i}.\mathrm{e}\:\:\:\:{d}\left(\mathrm{1000}\right)\:=\:\mathrm{1}\:,\:\:\:\:\:{d}\left(\mathrm{999}\right)\:=\:\mathrm{27} \\ $$$$\: \\ $$$$\:\mathrm{find}\:\mathrm{minimum}\:{k}\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\underset{{k}\:\mathrm{times}} {\underbrace{{d}\left({d}\left(…..}{d}}\left(\mathrm{5}^{\mathrm{10}^{\mathrm{100}} } \right)…..\right)\:\ll\:\mathrm{10}\right. \\ $$$$\: \\ $$

Commented by Rasheed.Sindhi last updated on 09/Sep/21

$${k}=\mathrm{3} \\ $$

Commented by talminator2856791 last updated on 09/Sep/21

$$\:\mathrm{wheres}\:\mathrm{the}\:\mathrm{proof}? \\ $$

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