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Category: Number Theory

1-2-3-4-5-14-15-by-M-A-

Question Number 164940 by amin96 last updated on 23/Jan/22 $$−−−−−−−−− \\ $$$$\mathrm{1}!−\mathrm{2}!+\mathrm{3}!−\mathrm{4}!+\mathrm{5}!−\ldots−\mathrm{14}!+\mathrm{15}!=? \\ $$$$ \\ $$$$−−−−−−−−−−\boldsymbol{{by}}\:\boldsymbol{{M}}.\boldsymbol{{A}} \\ $$ Commented by MJS_new last updated on 24/Jan/22…

Find-the-pricipal-and-ordinary-argument-of-z-i-2-2i-

Question Number 33463 by NECx last updated on 17/Apr/18 $${Find}\:{the}\:{pricipal}\:{and}\:{ordinary} \\ $$$${argument}\:{of}\:{z}=\frac{{i}}{−\mathrm{2}−\mathrm{2}{i}} \\ $$ Commented by abdo imad last updated on 17/Apr/18 $${z}\:=\:\frac{−{i}}{\mathrm{2}\left(\mathrm{1}+{i}\right)}\:=\:\frac{−{i}\left(\mathrm{1}−{i}\right)}{\mathrm{4}}\:=\:\frac{\mathrm{1}−{i}}{\mathrm{4}}\:\Rightarrow\mid{z}\mid\:=\frac{\mathrm{1}}{\mathrm{4}}\mid\mathrm{1}−{i}\mid\:=\frac{\sqrt{\mathrm{2}}}{\mathrm{4}} \\ $$$${z}\:=\frac{\mathrm{1}}{\mathrm{4}}\:−\frac{{i}}{\mathrm{4}}\:=\frac{\sqrt{\mathrm{2}}}{\mathrm{4}}\left(\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:−\frac{{i}}{\:\sqrt{\mathrm{2}}}\right)\:=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\:{e}^{−{i}\frac{\pi}{\mathrm{4}}}…

Prove-that-gcd-gcd-A-B-gcd-B-C-gcd-C-A-gcd-A-B-C-

Question Number 33369 by Rasheed.Sindhi last updated on 15/Apr/18 $$\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\mathrm{gcd}\left(\:\:\mathrm{gcd}\left(\mathrm{A},\mathrm{B}\right),\mathrm{gcd}\left(\mathrm{B},\mathrm{C}\right),\mathrm{gcd}\left(\mathrm{C},\mathrm{A}\right)\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{gcd}\left(\mathrm{A},\mathrm{B},\mathrm{C}\right) \\ $$ Answered by MJS last updated on 15/Apr/18 $$\mathrm{set}\:\mathrm{theory} \\…

Question-related-to-Q-33217-If-A-1-A-2-A-n-are-n-points-with-integer-coordinates-of-a-plane-such-that-every-triangle-whose-vertices-are-any-three-of-the-above-points-has-its-centroid-with-at-lea

Question Number 33323 by Rasheed.Sindhi last updated on 14/Apr/18 $$\mathrm{Question}\:\mathrm{related}\:\mathrm{to}\:\mathrm{Q}#\mathrm{33217} \\ $$$$\mathrm{If}\:\mathrm{A}_{\mathrm{1}} ,\mathrm{A}_{\mathrm{2}} ,…\mathrm{A}_{\mathrm{n}} \:\mathrm{are}\:\mathrm{n}\:\mathrm{points}\:\mathrm{with}\:\mathrm{integer} \\ $$$$\mathrm{coordinates}\:\mathrm{of}\:\mathrm{a}\:\mathrm{plane}\:\mathrm{such}\:\mathrm{that}\:\mathrm{every}\:\mathrm{triangle} \\ $$$$\mathrm{whose}\:\mathrm{vertices}\:\mathrm{are}\:\mathrm{any}\:\mathrm{three}\:\mathrm{of}\:\mathrm{the}\:\mathrm{above} \\ $$$$\mathrm{points}\:\mathrm{has}\:\mathrm{its}\:\mathrm{centroid}\:\mathrm{with}\:\mathrm{at}\:\mathrm{least}\:\mathrm{one} \\ $$$$\mathrm{non}-\mathrm{integer}\:\mathrm{coordinate}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{maximum} \\ $$$$\mathrm{possible}\:\mathrm{n}.\:\:\:…

Question-98596

Question Number 98596 by bemath last updated on 15/Jun/20 Commented by bobhans last updated on 15/Jun/20 $$\Rightarrow\left({x}^{\mathrm{3}} +{x}^{\mathrm{2}} −\mathrm{2}\right)+\left({x}^{\mathrm{2}} +\mathrm{3}{x}−\mathrm{4}\right){i}\:=\:\mathrm{0} \\ $$$$\left(\mathrm{1}\right)\:{real}\:{parts}\:\Rightarrow{x}^{\mathrm{3}} +{x}^{\mathrm{2}} −\mathrm{2}\:=\:\mathrm{0} \\…