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Category: Number Theory

Question-112575

Question Number 112575 by Aina Samuel Temidayo last updated on 08/Sep/20 Answered by Rasheed.Sindhi last updated on 09/Sep/20 $${x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} =\mathrm{1376} \\ $$$$\because\mathrm{4}\mid\mathrm{1376}\:\:\:\:\:\therefore\:\mathrm{4}\mid{x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} \\…

Question-112573

Question Number 112573 by Aina Samuel Temidayo last updated on 08/Sep/20 Answered by Rasheed.Sindhi last updated on 08/Sep/20 $${Let}\:{two}\:{numbers}\:{are}\:\mathrm{10}{a}+{b}\:\& \\ $$$$\mathrm{10}{c}+{d}\:{with}\:\mathrm{10}{a}+{b}>\mathrm{10}{c}+{d} \\ $$$${After}\:{written}\:{beside}: \\ $$$$\mathrm{100}\left(\mathrm{10}{a}+{b}\right)+\left(\mathrm{10}{c}+{d}\right)…

Let-K-be-the-product-of-all-factors-b-a-not-necessarily-distinct-where-a-and-b-are-integers-satisfying-1-a-b-10-Find-the-greatest-integer-n-such-that-2-n-divides-K-

Question Number 112535 by Aina Samuel Temidayo last updated on 08/Sep/20 $$\mathrm{Let}\:\mathrm{K}\:\mathrm{be}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{all}\:\mathrm{factors} \\ $$$$\left(\mathrm{b}−\mathrm{a}\right)\:\left(\mathrm{not}\:\mathrm{necessarily}\:\mathrm{distinct}\right) \\ $$$$\mathrm{where}\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{are}\:\mathrm{integers}\:\mathrm{satisfying} \\ $$$$\mathrm{1}\leqslant\mathrm{a}\leqslant\mathrm{b}\leqslant\mathrm{10}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{greatest} \\ $$$$\mathrm{integer}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that}\:\mathrm{2}^{\mathrm{n}} \:\mathrm{divides}\:\mathrm{K}. \\ $$$$ \\ $$…

Question-112465

Question Number 112465 by Aina Samuel Temidayo last updated on 08/Sep/20 Answered by floor(10²Eta[1]) last updated on 08/Sep/20 $$\mathrm{3}\mid\mathrm{3}.\mathrm{4}^{\mathrm{n}} +\mathrm{51}=\mathrm{3}\left(\mathrm{4}^{\mathrm{n}} +\mathrm{17}\right)\:\left(\mathrm{clearly}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{3}\right) \\ $$$$\mathrm{9}\mid\mathrm{3}.\mathrm{4}^{\mathrm{n}} +\mathrm{51}\Leftrightarrow\mathrm{3}\mid\mathrm{4}^{\mathrm{n}} +\mathrm{17}\Leftrightarrow\mathrm{3}\mid\left(\mathrm{4}^{\mathrm{n}}…

SOLVE-the-EQUATION-n-n-n-1-3-n-1-6-2016-

Question Number 112323 by Rasheed.Sindhi last updated on 07/Sep/20 $$\:_{\leqslant} ^{\geqslant} =\:\:\mathbb{SOLVE}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathcal{EQUATION}}_{} ^{\:_{\:\:\bullet} } \\ $$$$\:\:\:\boldsymbol{\mathrm{n}}−\lfloor\sqrt{\boldsymbol{\mathrm{n}}}\rfloor−\lfloor\sqrt[{\mathrm{3}}]{\boldsymbol{\mathrm{n}}}\rfloor+\lfloor\sqrt[{\mathrm{6}}]{\boldsymbol{\mathrm{n}}}\rfloor=\mathrm{2016} \\ $$$$ \\ $$ Commented by Rasheed.Sindhi last updated…

Prove-that-there-exist-4-distinct-positive-integers-such-that-each-integer-divides-the-sum-of-the-remaining-integers-

Question Number 46751 by Joel578 last updated on 31/Oct/18 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{there}\:\mathrm{exist}\:\mathrm{4}\:\mathrm{distinct}\:\mathrm{positive}\:\mathrm{integers} \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{each}\:\mathrm{integer}\:\mathrm{divides}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{remaining}\:\mathrm{integers}.\: \\ $$ Commented by MrW3 last updated on 31/Oct/18 $${n},\mathrm{2}{n},\mathrm{3}{n},\mathrm{6}{n}\:{with}\:{n}\in\mathbb{N} \\…