Question Number 226919 by Spillover last updated on 19/Dec/25
Answered by Ghisom_ last updated on 19/Dec/25
$$\underset{{n}} {\underbrace{\mathrm{6}…}}\mathrm{8}^{\mathrm{2}} =\underset{{n}} {\underbrace{\mathrm{4}…}}\mathrm{6}\underset{{n}} {\underbrace{\mathrm{2}…}}\mathrm{4} \\ $$$$\Rightarrow\:\underset{{digits}} {\sum}=\left(\mathrm{4}+\mathrm{2}\right){n}+\mathrm{6}+\mathrm{4}=\mathrm{6}{n}+\mathrm{10} \\ $$$${n}=\mathrm{10}^{\mathrm{5}} −\mathrm{1}\:\Rightarrow\:\underset{{digits}} {\sum}=\mathrm{600004} \\ $$
Commented by Spillover last updated on 19/Dec/25
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Answered by Raphael254 last updated on 19/Dec/25
$$ \\ $$$$\mathrm{68}^{\mathrm{2}} \:=\:\mathrm{4624},\:\mathrm{668}^{\mathrm{2}} \:=\:\mathrm{446224}, \\ $$$$\mathrm{6668}^{\mathrm{2}} \:=\:\mathrm{44462224} \\ $$$$ \\ $$$$\left(\mathrm{6666666}…\mathrm{68}\right)^{\mathrm{2}} \:=\:\mathrm{444}…\mathrm{6}…\mathrm{222}…\mathrm{4} \\ $$$$ \\ $$$$\left(\underset{{i}=\mathrm{1}} {\overset{\mathrm{10}^{\mathrm{5}} −\mathrm{1}} {\sum}}\:\left(\mathrm{4}\:+\:\mathrm{2}\right)\right)\:+\:\left(\mathrm{6}\:+\:\mathrm{4}\right) \\ $$$$=\:\left(\underset{{i}=\mathrm{1}} {\overset{\mathrm{10}^{\mathrm{5}} −\mathrm{1}} {\sum}}\:\mathrm{6}\right)\:+\:\mathrm{10} \\ $$$$=\:\mathrm{6}×\left(\mathrm{10}^{\mathrm{5}} −\mathrm{1}\right)+\mathrm{10} \\ $$$$=\:\mathrm{6}×\mathrm{10}^{\mathrm{5}} −\mathrm{6}+\mathrm{10} \\ $$$$=\:\mathrm{6}×\mathrm{10}^{\mathrm{5}} +\mathrm{4} \\ $$$$=\:\mathrm{600000}+\mathrm{4} \\ $$$$=\:\mathrm{600004} \\ $$
Commented by Spillover last updated on 19/Dec/25
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Answered by HeMath last updated on 19/Dec/25
$$\:\underbrace{\:\underset{\mathrm{10}^{\mathrm{5}} \:\mathrm{digits}} {\mathrm{6666}…\mathrm{68}^{\mathrm{2}} \:}}\:=\:\left(\mathrm{6}.\mathrm{666}\:×\:\mathrm{10}^{\mathrm{5}} \right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{44}.\mathrm{4444}\:×\:\mathrm{10}^{\mathrm{10}\:} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{4}.\mathrm{44}\:×\:\mathrm{10}^{\mathrm{11}\:} \Rightarrow\:\mathrm{10}^{\mathrm{11}} \:\mathrm{digits}\:# \\ $$
Commented by Spillover last updated on 19/Dec/25
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