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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=3

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

Commented by talminator2856791 last updated on 09/Sep/21

 wheres the proof?

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

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