Question Number 149568 by mathdanisur last updated on 06/Aug/21
$${solve}\:{the}\:{equation}: \\ $$$$\mathrm{20}{z}\left[{z}\right]\:-\:\mathrm{21}\left\{{z}\right\}\:=\:\mathrm{2021} \\ $$$${where}\:\left\{\ast\right\}\:{is}\:{GIF}\:\:{and}\:\:\left\{{z}\right\}\:=\:{z}\:-\:\left[{z}\right] \\ $$
Answered by Olaf_Thorendsen last updated on 06/Aug/21
$$\mathrm{20}{z}\left[{z}\right]−\mathrm{21}\left\{{z}\right\}\:=\:\mathrm{2021}\:\:\:\:\left(\mathrm{1}\right) \\ $$$$\mathrm{20}\left(\left[{z}\right]+\left\{{z}\right\}\right)\left[{z}\right]−\mathrm{21}\left\{{z}\right\}\:=\:\mathrm{2021} \\ $$$$\mathrm{Let}\:{n}\:=\:\left[{z}\right]\:\mathrm{and}\:{q}\:=\:\left\{{z}\right\}\:\left(\mathrm{0}\leqslant{q}<\mathrm{1}\right\} \\ $$$$\mathrm{20}{n}^{\mathrm{2}} +\mathrm{20}{qn}−\mathrm{21}{q}\:=\:\mathrm{2021} \\ $$$${q}\:=\:\frac{\mathrm{2021}−\mathrm{20}{n}^{\mathrm{2}} }{\mathrm{20}{n}−\mathrm{21}}\:\:\:\:\left(\mathrm{2}\right) \\ $$$$\mathrm{By}\:\mathrm{plotting}\:{q}\:\mathrm{against}\:{n},\:\mathrm{we}\:\mathrm{verify} \\ $$$$\mathrm{that}\:\mathrm{0}\leqslant{q}<\mathrm{1}\:\mathrm{for}\:{n}\:=\:\mathrm{10}\:\mathrm{only} \\ $$$$ \\ $$$$\bullet\:\mathrm{if}\:\left[{z}\right]\:=\:\mathrm{10} \\ $$$$\left(\mathrm{2}\right)\::\:\left\{{z}\right\}\:=\:\frac{\mathrm{2021}−\mathrm{20}×\mathrm{100}}{\mathrm{200}−\mathrm{21}}\:=\:\frac{\mathrm{21}}{\mathrm{179}} \\ $$$$\Rightarrow\:{z}\:=\:\left[{x}\right]+\left\{{z}\right\}\:=\:\mathrm{10}+\frac{\mathrm{21}}{\mathrm{179}}\:=\:\frac{\mathrm{1811}}{\mathrm{179}} \\ $$$$ \\ $$$$\mathrm{We}\:\mathrm{verify}\:\mathrm{in}\:\left(\mathrm{1}\right)\:\mathrm{that}\:: \\ $$$$\mathrm{20}.\frac{\mathrm{1811}}{\mathrm{179}}.\mathrm{10}−\mathrm{21}.\frac{\mathrm{21}}{\mathrm{179}}\:=\:\mathrm{2021} \\ $$$$ \\ $$$$\boldsymbol{{z}}\:=\:\frac{\mathrm{1811}}{\mathrm{179}} \\ $$
Commented by mathdanisur last updated on 07/Aug/21
$$\boldsymbol{\mathrm{S}}\mathrm{er},\:\mathrm{thank}\:\mathrm{you}\:\mathrm{cool} \\ $$