Question Number 162674 by Mathematification last updated on 31/Dec/21 Answered by mindispower last updated on 31/Dec/21 $${d}\mid\mathrm{2}\left(\mathrm{3}{n}+\mathrm{1}\right)−\mathrm{3}\left(\mathrm{2}{n}+\mathrm{1}\right)\Rightarrow{d}\mid−\mathrm{1} \\ $$$${d}=\mathrm{1} \\ $$$${or}\:{use}\:{bizou}\:{identities} \\ $$$$−\mathrm{2}\left(\mathrm{3}{n}+\mathrm{1}\right)+\mathrm{3}\left(\mathrm{2}{n}+\mathrm{1}\right)=\mathrm{1} \\ $$$$\Rightarrow\left(\mathrm{2}{n}+\mathrm{1}\right),\left(\mathrm{3}{n}+\mathrm{1}\right)\:{are}\:{prime}…
Question Number 31579 by Joel578 last updated on 10/Mar/18 $$\mathrm{Let}\:{a}\:\mathrm{and}\:{b}\:\mathrm{be}\:\mathrm{an}\:\mathrm{integer}\:\mathrm{part}\:\mathrm{and}\:\mathrm{a}\:\mathrm{decimal} \\ $$$$\mathrm{fraction}\:\mathrm{of}\:\sqrt{\mathrm{7}},\:\mathrm{respectively}.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{integer} \\ $$$$\mathrm{part}\:\mathrm{of}\:\frac{{a}}{{b}}\:\mathrm{is}? \\ $$ Answered by Tinkutara last updated on 10/Mar/18 $${a}=\mathrm{2} \\…
Question Number 96928 by bobhans last updated on 05/Jun/20 $$\mathrm{69}{x}\:\equiv\:\mathrm{1}\:\left(\mathrm{mod}\:\mathrm{31}\right)\: \\ $$$$\mathrm{solve}\:\mathrm{for}\:{x} \\ $$ Commented by RAMANA last updated on 05/Jun/20 $${send}\:{the}\:{answer}\:{please} \\ $$ Answered…
Question Number 31125 by Joel578 last updated on 02/Mar/18 $$\mathrm{Let}\:{n}\:\mathrm{be}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{integer}.\:\mathrm{Then}\:{x}^{\mathrm{2}} \:+\:\mathrm{1}\: \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{factor}\:\mathrm{of}\:\left({x}^{\mathrm{4}} \:+\:\mathrm{3}\right)^{{n}} \:−\:\left[\left({x}^{\mathrm{2}} \:+\:\mathrm{3}\right)\left({x}^{\mathrm{2}} \:−\:\mathrm{1}\right)\right]^{{n}} \\ $$$$\mathrm{for}\:… \\ $$$$\left(\mathrm{A}\right)\:\mathrm{All}\:{n} \\ $$$$\left(\mathrm{B}\right)\:\mathrm{Odd}\:{n} \\ $$$$\left(\mathrm{C}\right)\:\mathrm{Even}\:{n}…
Question Number 162169 by Rasheed.Sindhi last updated on 27/Dec/21 $$ \\ $$$$\:\:\:\:\:\:\:\begin{array}{|c|}{\overset{\bullet} {\:\:\:\:\:\begin{array}{|c|}{\:\:\:\underset{{x}=?,{y}=?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:} {\overset{{x},{y}\in\mathbb{Z}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:} {{x}+{y}+{x}^{\mathrm{2}} {y}^{\mathrm{2}} =\mathrm{586}}}\:\:}\\\hline\end{array}_{} ^{} }\:\:\:\:}\\\hline\end{array} \\ $$$$ \\ $$ Answered by…
Question Number 96558 by bobhans last updated on 02/Jun/20 $$\mathrm{Let}\:\mathrm{m}\:\mathrm{and}\:\mathrm{n}\:\mathrm{be}\:\mathrm{two}\:\mathrm{positive}\:\mathrm{integers}\: \\ $$$$\mathrm{satisfy}\:\frac{\mathrm{m}}{\mathrm{n}}\:=\:\frac{\mathrm{1}}{\mathrm{10}×\mathrm{12}}+\frac{\mathrm{1}}{\mathrm{12}×\mathrm{14}}+\frac{\mathrm{1}}{\mathrm{14}×\mathrm{16}}+…+\frac{\mathrm{1}}{\mathrm{2012}×\mathrm{2014}} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{m}+\mathrm{n}\: \\ $$ Answered by john santu last updated on…
Question Number 96437 by bemath last updated on 01/Jun/20 Answered by bobhans last updated on 01/Jun/20 $$\boldsymbol{{xy}}−\mathrm{3}\boldsymbol{{x}}+\mathrm{5}\boldsymbol{{y}}−\mathrm{15}\:=\:−\mathrm{15} \\ $$$$\left(\boldsymbol{{x}}+\mathrm{5}\right)\left(\boldsymbol{{y}}−\mathrm{3}\right)\:=\:−\mathrm{15} \\ $$$$\left(\mathrm{1}\right)\:\boldsymbol{{x}}+\mathrm{5}\:=\:\pm\mathrm{1};\:\mathrm{y}−\mathrm{3}\:=\:\mp\mathrm{15} \\ $$$$\left(\mathrm{2}\right){x}+\mathrm{5}\:=\:\pm\:\mathrm{3}\:;\:\mathrm{y}−\mathrm{3}\:=\:\mp\:\mathrm{5}\: \\ $$$$\left(\mathrm{3}\right)\:{x}+\mathrm{5}\:=\:\pm\mathrm{5}\:;\:\mathrm{y}−\mathrm{3}\:=\:\mp\:\mathrm{3}…
Question Number 30897 by usman katyar last updated on 28/Feb/18 $$\left(\mathrm{44}{x}+\mathrm{28}{y}\right)\mathrm{2} \\ $$ Commented by rahul 19 last updated on 28/Feb/18 $$? \\ $$$$\mathrm{88}{x}+\mathrm{56}{y}\:. \\…
Question Number 161860 by Rasheed.Sindhi last updated on 23/Dec/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Simplify} \\ $$$$\frac{\mathrm{1}^{\mathrm{2}} \centerdot\mathrm{2}!+\mathrm{2}^{\mathrm{2}} \centerdot\mathrm{3}!+\mathrm{3}^{\mathrm{2}} \centerdot\mathrm{4}!+\centerdot\centerdot\centerdot+{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)!−\mathrm{2}}{\left({n}+\mathrm{1}\right)!} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{to} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{n}^{\mathrm{2}} +\mathrm{n}−\mathrm{2} \\ $$ Answered by…
Question Number 161843 by Rasheed.Sindhi last updated on 23/Dec/21 $${Q}#\mathrm{161744}\:{reposted}\:{with}\:{some}\:{change}. \\ $$$$\mathrm{Solve}\:\mathrm{for}\:\boldsymbol{\mathrm{integer}}\:\mathrm{numbers}: \\ $$$$\frac{\mathrm{x}}{\mathrm{y}}\:+\:\frac{\mathrm{5}}{\mathrm{x}}\:+\:\frac{\mathrm{y}\:-\:\mathrm{5}}{\mathrm{5}}\:=\:\frac{\mathrm{y}\:+\:\mathrm{x}}{\mathrm{y}\:+\:\mathrm{5}}\:+\:\frac{\mathrm{5}\:+\:\mathrm{y}}{\mathrm{5}\:+\:\mathrm{x}} \\ $$ Commented by malwan last updated on 23/Dec/21 $${x}=\mathrm{0}\:{or}\:{x}=\mathrm{5} \\…