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Given-function-f-4567-321567-567-321-888-f-32156-12062-156-120-276-find-the-value-of-f-20-22-21-

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Question Number 198447 by cortano12 last updated on 20/Oct/23
Given function f(4567,321567)= 567+321=888. f(32156,12062)= 156+120=276 find the value of f(((20^(22) )/(21)) ).
$$\:\:\mathrm{Given}\:\mathrm{function}\: \\ $$$$\:\:\mathrm{f}\left(\mathrm{4567},\mathrm{321567}\right)=\:\mathrm{567}+\mathrm{321}=\mathrm{888}. \\ $$$$\:\:\mathrm{f}\left(\mathrm{32156},\mathrm{12062}\right)=\:\mathrm{156}+\mathrm{120}=\mathrm{276} \\ $$$$\:\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\:\:\:\:\:\mathrm{f}\left(\frac{\mathrm{20}^{\mathrm{22}} }{\mathrm{21}}\:\right). \\ $$
Commented by mr W last updated on 20/Oct/23
the given function is f(x,y), but you are asking f(a)=? it makes no sense to me.
$${the}\:{given}\:{function}\:{is}\:{f}\left({x},{y}\right),\:{but}\:{you}\: \\ $$$${are}\:{asking}\:{f}\left({a}\right)=? \\ $$$${it}\:{makes}\:{no}\:{sense}\:{to}\:{me}. \\ $$
Commented by Rasheed.Sindhi last updated on 20/Oct/23
sir, I think comma is decimal separater here.
$$\boldsymbol{{sir}},\:{I}\:{think}\:{comma}\:{is}\: \\ $$$${decimal}\:{separater}\:{here}. \\ $$
Commented by cortano12 last updated on 20/Oct/23
the questions mean if f(((157683)/(321)))=f(491,785047)=491+785
$$\mathrm{the}\:\mathrm{questions}\:\mathrm{mean}\:\mathrm{if}\: \\ $$$$\:\mathrm{f}\left(\frac{\mathrm{157683}}{\mathrm{321}}\right)=\mathrm{f}\left(\mathrm{491},\mathrm{785047}\right)=\mathrm{491}+\mathrm{785} \\ $$
Commented by mr W last updated on 20/Oct/23
ok. “,”=“.”
$${ok}.\:“,''=“.'' \\ $$
Commented by Rasheed.Sindhi last updated on 20/Oct/23
f(((157683)/(321)))=f(491,785047)=491+785 But ((157683)/(321))=491.224299≠491,785047
$$\:\mathrm{f}\left(\frac{\mathrm{157683}}{\mathrm{321}}\right)=\mathrm{f}\left(\mathrm{491},\mathrm{785047}\right)=\mathrm{491}+\mathrm{785} \\ $$$$ \\ $$$${But}\:\frac{\mathrm{157683}}{\mathrm{321}}=\mathrm{491}.\mathrm{224299}\neq\mathrm{491},\mathrm{785047} \\ $$
Answered by Rasheed.Sindhi last updated on 20/Oct/23
f(4567,321567)= 567+321=888 f(32156,12062)= 156+120=276 f(((20^(22) )/(21)) )=? f(((20^(22) )/(21)) )=⌊((20^(22) )/(21))⌋mod 1000 +⌊((20^(22) )/(21))×1000⌋mod 1000 ∵ ((20^(22) )/(21)) = (2^(22) /(21))×10^(22) =⌊(2^(22) /(21))×10^(22) ⌋mod10^3 +⌊(2^(22) /(21))×10^(25) ⌋mod10^3 =619+47=666 Continue
$$\:\:\mathrm{f}\left(\mathrm{4567},\mathrm{321567}\right)=\:\mathrm{567}+\mathrm{321}=\mathrm{888} \\ $$$$\:\:\mathrm{f}\left(\mathrm{32156},\mathrm{12062}\right)=\:\mathrm{156}+\mathrm{120}=\mathrm{276} \\ $$$$\:\:\:\:\:\mathrm{f}\left(\frac{\mathrm{20}^{\mathrm{22}} }{\mathrm{21}}\:\right)=? \\ $$$$\mathrm{f}\left(\frac{\mathrm{20}^{\mathrm{22}} }{\mathrm{21}}\:\right)=\lfloor\frac{\mathrm{20}^{\mathrm{22}} }{\mathrm{21}}\rfloor{mod}\:\mathrm{1000} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\lfloor\frac{\mathrm{20}^{\mathrm{22}} }{\mathrm{21}}×\mathrm{1000}\rfloor{mod}\:\mathrm{1000} \\ $$$$\:\:\:\:\:\:\because\:\:\:\:\:\:\frac{\mathrm{20}^{\mathrm{22}} }{\mathrm{21}}\:=\:\frac{\mathrm{2}^{\mathrm{22}} }{\mathrm{21}}×\mathrm{10}^{\mathrm{22}} \\ $$$$=\lfloor\frac{\mathrm{2}^{\mathrm{22}} }{\mathrm{21}}×\mathrm{10}^{\mathrm{22}} \rfloor{mod}\mathrm{10}^{\mathrm{3}} +\lfloor\frac{\mathrm{2}^{\mathrm{22}} }{\mathrm{21}}×\mathrm{10}^{\mathrm{25}} \rfloor{mod}\mathrm{10}^{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\mathrm{619}+\mathrm{47}=\mathrm{666} \\ $$$${Continue} \\ $$
Answered by MM42 last updated on 20/Oct/23
20^(22) ≡^(21) 1⇒20^(22) =21k+1 ; k=abc...619 ⇒((20^(22) )/(21))=((21×abc..619+1)/(21))=abc..619+0.476.. ⇒f(((20^(22) )/(21)))=619+47=666 ✓
$$\mathrm{20}^{\mathrm{22}} \:\overset{\mathrm{21}} {\equiv}\mathrm{1}\Rightarrow\mathrm{20}^{\mathrm{22}} =\mathrm{21}{k}+\mathrm{1}\:;\:\:{k}={abc}…\mathrm{619} \\ $$$$\Rightarrow\frac{\mathrm{20}^{\mathrm{22}} }{\mathrm{21}}=\frac{\mathrm{21}×{abc}..\mathrm{619}+\mathrm{1}}{\mathrm{21}}={abc}..\mathrm{619}+\mathrm{0}.\mathrm{476}.. \\ $$$$\Rightarrow{f}\left(\frac{\mathrm{20}^{\mathrm{22}} }{\mathrm{21}}\right)=\mathrm{619}+\mathrm{47}=\mathrm{666}\:\checkmark \\ $$$$ \\ $$
Commented by cortano12 last updated on 20/Oct/23
how to get k=abc...619?
$$\mathrm{how}\:\mathrm{to}\:\mathrm{get}\:\mathrm{k}=\mathrm{abc}…\mathrm{619}? \\ $$
Commented by MM42 last updated on 20/Oct/23
20^(22) −1=a_1 ..a_n 9...999=21k if k=abc...xyz ⇒ 21×...xyz=...999 ⇒z=9⇒21×...xy9=...999 y+8=9⇒y=1 ⇒21×...x19=...999 x+3=9⇒x=6 ⇒k=abc...619 ✓
$$\mathrm{20}^{\mathrm{22}} −\mathrm{1}={a}_{\mathrm{1}} ..{a}_{{n}} \mathrm{9}…\mathrm{999}=\mathrm{21}{k} \\ $$$${if}\:\:{k}={abc}…{xyz}\:\Rightarrow\:\mathrm{21}×…{xyz}=…\mathrm{999} \\ $$$$\Rightarrow{z}=\mathrm{9}\Rightarrow\mathrm{21}×…{xy}\mathrm{9}=…\mathrm{999} \\ $$$$\underbrace{ } \\ $$$${y}+\mathrm{8}=\mathrm{9}\Rightarrow{y}=\mathrm{1} \\ $$$$\Rightarrow\mathrm{21}×…{x}\mathrm{19}=…\mathrm{999} \\ $$$${x}+\mathrm{3}=\mathrm{9}\Rightarrow{x}=\mathrm{6} \\ $$$$\Rightarrow{k}={abc}…\mathrm{619}\:\:\checkmark\: \\ $$$$\underbrace{ } \\ $$

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