A-20-digit-decimal-number-has-been-converted-into-octal-system-Say-it-has-n-digits-What-can-be-minimum-and-maximum-possible-values-of-n-

Question Number 35071 by Rasheed.Sindhi last updated on 15/May/18

A 20 - digit decimal number has been

converted into octal system . Say it has

n digits . What can be minimum and

maximum possible values of n ?

Answered by candre last updated on 15/May/18

m = l α^{k} + d; 0 < l ⩽ α - 1 \land 0 ⩽ d < α^{k} \land (l, d, k, α) \in N^{4} \land α > 1

m = (l + d α^{- k}) α^{k}

\log_{α} m = \log_{α} [(l + d α^{- k}) α^{k}]

= \log_{α} α^{k} + \log_{α} (l + d α^{- k})

= k \log_{α} α + \log_{α} (l + d α^{- k})

= k + \log_{α} (l + d α^{- k})

0 ⩽ d < α^{k} \Rightarrow 0 ⩽ d α^{- k} < 1

0 < l + d α^{- k} < α

\log_{α} (l + d α^{- k}) < 1

α^{k} ⩽ m < α^{k + 1} \Rightarrow k ⩽ \log_{α} m < k + 1

m = a_{k} \dots a_{0} \Rightarrow k - 0 + 1 = k + 1

m i n \Rightarrow n u m = 10^{19}

m i n = ⌊ \log_{8} 10^{19} ⌋ + 1 = ⌊ {19 \log}_{8} 10 ⌋ + 1 = 22

m a x \Rightarrow n u m = 10^{20} - 1

m a x = ⌊ \log_{8} (10^{20} - 1) ⌋ + 1 = 23

Commented by Rasheed.Sindhi last updated on 15/May/18

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