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Question Number 158711 by HongKing last updated on 07/Nov/21

Commented by Rasheed.Sindhi last updated on 08/Nov/21

1000000

$$\mathrm{1000000} \\ $$

Commented by HongKing last updated on 08/Nov/21

how my dear Sir...

$$\mathrm{how}\:\mathrm{my}\:\mathrm{dear}\:\boldsymbol{\mathrm{S}}\mathrm{ir}... \\ $$

Answered by ajfour last updated on 08/Nov/21

N=n^2 =m^3 =20k=2^2 ×5×k_(min) primes cannot be compensated N=2^6 ×5^6 =10^6 =1million

$${N}={n}^{\mathrm{2}} ={m}^{\mathrm{3}} =\mathrm{20}{k}=\mathrm{2}^{\mathrm{2}} ×\mathrm{5}×{k}_{{min}} \\ $$$${primes}\:{cannot}\:{be}\:{compensated} \\ $$$${N}=\mathrm{2}^{\mathrm{6}} ×\mathrm{5}^{\mathrm{6}} =\mathrm{10}^{\mathrm{6}} =\mathrm{1}{million} \\ $$

Commented by HongKing last updated on 08/Nov/21

very nice dear Ser thank you

$$\mathrm{very}\:\mathrm{nice}\:\mathrm{dear}\:\mathrm{Ser}\:\mathrm{thank}\:\mathrm{you} \\ $$

Answered by Rasheed.Sindhi last updated on 08/Nov/21

Required number: A=2^(m≥2) .5^(n≥1) [∵ 20(=2^2 .5^1 ) ∣ A ]..........(i) A=2^(k.LCM(2,3)) .5^(k.LCM(2,3)) .............(ii) [∵ A is same time perfect square (□^2 ) & perfect cube(△^3 )] (i) & (ii):A=2^(1×6) .5^(1×6) =1000000 [A is the least such number so k=1]

$${Required}\:{number}: \\ $$$${A}=\mathrm{2}^{{m}\geqslant\mathrm{2}} .\mathrm{5}^{{n}\geqslant\mathrm{1}} \:\left[\because\:\mathrm{20}\left(=\mathrm{2}^{\mathrm{2}} .\mathrm{5}^{\mathrm{1}} \right)\:\mid\:{A}\:\right]..........\left(\mathrm{i}\right) \\ $$$${A}=\mathrm{2}^{\mathrm{k}.\mathrm{LCM}\left(\mathrm{2},\mathrm{3}\right)} .\mathrm{5}^{\mathrm{k}.\mathrm{LCM}\left(\mathrm{2},\mathrm{3}\right)} .............\left(\mathrm{ii}\right) \\ $$$$\left[\because\:{A}\:{is}\:{same}\:{time}\:{perfect}\:{square}\right. \\ $$$$\left.\left(\Box^{\mathrm{2}} \right)\:\&\:{perfect}\:{cube}\left(\bigtriangleup^{\mathrm{3}} \right)\right] \\ $$$$\left(\mathrm{i}\right)\:\&\:\left(\mathrm{ii}\right):{A}=\mathrm{2}^{\mathrm{1}×\mathrm{6}} .\mathrm{5}^{\mathrm{1}×\mathrm{6}} =\mathrm{1000000} \\ $$$$\left[{A}\:{is}\:{the}\:{least}\:{such}\:{number}\:{so}\:\mathrm{k}=\mathrm{1}\right] \\ $$

Commented by HongKing last updated on 08/Nov/21

very nice dear Ser thank you

$$\mathrm{very}\:\mathrm{nice}\:\mathrm{dear}\:\mathrm{Ser}\:\mathrm{thank}\:\mathrm{you} \\ $$

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