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a-2-b-1-1999-how-many-natural-solutions-of-the-equation-a-b-have-




Question Number 199133 by hardmath last updated on 28/Oct/23
a^2 b − 1 = 1999  how many natural solutions of the  equation (a,b) have?
$$\mathrm{a}^{\mathrm{2}} \mathrm{b}\:−\:\mathrm{1}\:=\:\mathrm{1999} \\ $$$$\mathrm{how}\:\mathrm{many}\:\mathrm{natural}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\left(\mathrm{a},\mathrm{b}\right)\:\mathrm{have}? \\ $$
Answered by Rasheed.Sindhi last updated on 28/Oct/23
a^2 b − 1 = 1999  a^2 b=2000=2^4 .5^3   Possibilities for a^2   a^2 =1^(2k)  , 2^2 , 2^4 , 5^2 , 2^2 .5^2  , 2^4 .5^2   Six natural soutions.
$$\mathrm{a}^{\mathrm{2}} \mathrm{b}\:−\:\mathrm{1}\:=\:\mathrm{1999} \\ $$$$\mathrm{a}^{\mathrm{2}} \mathrm{b}=\mathrm{2000}=\mathrm{2}^{\mathrm{4}} .\mathrm{5}^{\mathrm{3}} \\ $$$${Possibilities}\:{for}\:\mathrm{a}^{\mathrm{2}} \\ $$$$\mathrm{a}^{\mathrm{2}} =\mathrm{1}^{\mathrm{2}{k}} \:,\:\mathrm{2}^{\mathrm{2}} ,\:\mathrm{2}^{\mathrm{4}} ,\:\mathrm{5}^{\mathrm{2}} ,\:\mathrm{2}^{\mathrm{2}} .\mathrm{5}^{\mathrm{2}} \:,\:\mathrm{2}^{\mathrm{4}} .\mathrm{5}^{\mathrm{2}} \\ $$$${Six}\:{natural}\:{soutions}. \\ $$
Commented by hardmath last updated on 28/Oct/23
thank you professor
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{professor} \\ $$

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