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Olympiad workbook
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IOQM
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Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

For any real number t , let [t] denote the largest integer ≤ t ....

For any real number t , let [t][\mathrm{t}] denote the largest integer ≤t\leq \mathrm{t}. Suppose that N is the greatest integer such that [[[N]]=4[\sqrt{[\sqrt{[\sqrt{\mathrm{N}}]}]}=4. Find the sum of digit of N .

Correct answer

24

Step-by-step solution

[[N]]=44≤[N]<516≤[[N]]<2516≤[N]<25256≤[N]<625256≤N<625(256)2≤N<(625)2Nmax. =390624 sum of digit =24 Ans. \begin{aligned} & {[\sqrt{[\sqrt{N}]}]=4} \\ & 4 \leq \sqrt{[\sqrt{\sqrt{N}}]} < 5 \\ & 16 \leq[\sqrt{[\sqrt{N}]}] < 25 \\ & 16 \leq \sqrt{[\sqrt{N}]} < 25 \\ & 256 \leq[\sqrt{N}] < 625 \\ & 256 \leq \sqrt{N} < 625 \\ & (256)^2 \leq N < (625)^2 \\ & N_{\text {max. }}=390624 \\ & \text { sum of digit }=24 \text { Ans. }\end{aligned}
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