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Category: Number Theory

1-Find-all-natural-pairs-of-integers-x-y-such-that-x-3-y-3-xy-61-2-Find-gcd-x-4-x-3-x-3-x-

Question Number 104100 by bramlex last updated on 19/Jul/20 $$\left(\mathrm{1}\right)\mathbb{F}{ind}\:{all}\:{natural}\:{pairs}\:{of} \\ $$$${integers}\:\left({x},{y}\right)\:{such}\:{that}\:{x}^{\mathrm{3}} −{y}^{\mathrm{3}} ={xy}+\mathrm{61}. \\ $$$$\left(\mathrm{2}\right)\mathbb{F}{ind}\:{gcd}\left({x}^{\mathrm{4}} −{x}^{\mathrm{3}} ,\:{x}^{\mathrm{3}} −{x}\right)\: \\ $$ Answered by bemath last…

1-Evaluate-the-sum-2-0-3-2-1-3-2-2-3-2-1000-3-2-find-2-98-mod-33-

Question Number 104093 by bramlex last updated on 19/Jul/20 $$\left(\mathrm{1}\right){Evaluate}\:{the}\:{sum}\:\lfloor\frac{\mathrm{2}^{\mathrm{0}} }{\mathrm{3}}\rfloor+\lfloor\frac{\mathrm{2}^{\mathrm{1}} }{\mathrm{3}}\rfloor+\lfloor\frac{\mathrm{2}^{\mathrm{2}} }{\mathrm{3}}\rfloor+…+\lfloor\frac{\mathrm{2}^{\mathrm{1000}} }{\mathrm{3}}\rfloor \\ $$$$\left(\mathrm{2}\right)\:{find}\:\mathrm{2}^{\mathrm{98}} \:\left({mod}\:\mathrm{33}\right)\: \\ $$ Answered by JDamian last updated on…