Category: Number Theory

Question Number 62554 by Rasheed.Sindhi last updated on 22/Jun/19 $$Ifa\mathrm{\&}baretwonaturalnumbers$$$$thenthefollowingrelationholds$$$$always.$$$$a\times b=\gcd (a,b)\times \mathrm{lcm}(a,b)$$$${}^{\bullet }Ifa,b\mathrm{\&}carethreenaturalnumbers$$$$whatrelation\left(anologoustotheabove\right)$$$$holdsbetweennumbersandtheir$$$$\:\:\:\mathrm{gcd}\:\&\:\mathrm{lcm}?…

Question Number 127940 by bramlexs22 last updated on 03/Jan/21 $$\{\begin{array}{l}3\mathrm{x}=1\left(\mathrm{mod}4\right)\\ 4\mathrm{x}=3\left(\mathrm{mod}5\right)\\ 5\mathrm{x}=7\left(\mathrm{mod}11\right)\end{array}$$ Answered by floor(10²Eta[1]) last updated on 03/Jan/21 $$3\mathrm{x}\equiv 1\left(\mathrm{mod}4\right)\Rightarrow \mathrm{x}\equiv 3\left(\mathrm{mod}4\right)$$$$\mathrm{x}=4\mathrm{a}+3$$$$\mathrm{4}\left(\mathrm{4a}+\mathrm{3}\right)\equiv\mathrm{3}\left(\mathrm{mod}\:\mathrm{5}\right) \…

Question Number 62242 by Rasheed.Sindhi last updated on 18/Jun/19 $$\mathrm{Find}\mathrm{out}\mathrm{x},\mathrm{y}\mathrm{such}\mathrm{that}$$$$\mathrm{lcm}(\mathrm{x},\mathrm{y})=180\wedge \gcd (\mathrm{x},\mathrm{y})=45$$ Commented by maxmathsup by imad last updated on 19/Jun/19 $$\Delta\left({x},{y}\right)\:=\mathrm{45}\:{and}\:{M}\left({x},{y}\right)=\mathrm{180}\:\Rightarrow{x}\:=\mathrm{45}\:{u}\:\:{and}\:{y}\:=\mathrm{45}\:{v}\:{with}\:\Delta\left({u},{v}\right)=\mathrm{1} \…

Question Number 127610 by pticantor last updated on 31/Dec/20 $$\mathit{Z}=1+(1+i)\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }$$$$\mathit{a}\mathit{r}\mathit{g}\left(\mathit{z}\right)=?$$ Answered by MJS_new last updated on 31/Dec/20 $$\arg (1+\cos \theta +\mathrm{i}\cos \theta )=$$$$=\frac{\pi}{\mathrm{2}}\mathrm{sign}\:\left(\mathrm{cos}\:\theta\right)\:−\mathrm{arctan}\:\frac{\mathrm{1}+\mathrm{cos}\:\theta}{\mathrm{cos}\:\theta} \…

Question Number 127111 by benjo_mathlover last updated on 26/Dec/20 $$findthevalueofxsuchthat$$$$\{\begin{array}{l}x=2\left(mod5\right)\\ x=3\left(mod8\right)\\ x=2\left(mod3\right)\end{array}$$ Answered by liberty last updated on 04/Jan/21 $$given\{\begin{array}{l}x=2\left(mod5\right)\dots \left(i\right)\\ x=3\left(mod8\right)\dots \left(ii\right)\\ x=2\left(mod3\right)\dots \left(iii\right)\end{array}$$$${for}\left({i}\right)\:\Rightarrow\:\mathrm{24}{a}\:\equiv\:\mathrm{2}\:\left({mod}\:\mathrm{5}\right)\: \…