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Category: Number Theory

Simplify-1-2-2-2-2-3-3-2-4-n-2-n-1-2-n-1-to-n-2-n-2-

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…

Q-161744-reposted-with-some-change-Solve-for-integer-numbers-x-y-5-x-y-5-5-y-x-y-5-5-y-5-x-

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} \\…

Given-that-LCM-A-B-C-252-LCM-A-B-36-amp-LCM-A-C-63-then-LCM-B-C-Pl-determine-all-possible-answers-

Question Number 30687 by Rasheed.Sindhi last updated on 24/Feb/18 $$\mathrm{Given}\:\mathrm{that}\:\mathrm{LCM}\left(\mathrm{A},\mathrm{B},\mathrm{C}\right)=\mathrm{252} \\ $$$$\mathrm{LCM}\left(\mathrm{A},\mathrm{B}\right)=\mathrm{36}\:\&\:\mathrm{LCM}\left(\mathrm{A},\mathrm{C}\right)=\mathrm{63}; \\ $$$$\mathrm{then}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{LCM}\left(\mathrm{B},\mathrm{C}\right)=? \\ $$$$\mathrm{Pl}\:\mathrm{determine}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{answers}. \\ $$ Answered by MJS last updated…

n-1-1-n-1-n-2n-1-

Question Number 161622 by amin96 last updated on 20/Dec/21 $$\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{\boldsymbol{{n}}+\mathrm{1}} }{\boldsymbol{{n}}\left(\mathrm{2}\boldsymbol{{n}}+\mathrm{1}\right)}=? \\ $$ Answered by mathmax by abdo last updated on 20/Dec/21 $$\frac{\mathrm{S}}{\mathrm{2}}=−\sum_{\mathrm{n}=\mathrm{1}}…

PROVE-that-the-numbers-of-types-4k-2-amp-4k-3-are-NOT-perfect-s-

Question Number 161528 by Rasheed.Sindhi last updated on 19/Dec/21 $$\mathrm{PROVE}\:\mathrm{that}\:\mathrm{the}\:\mathrm{numbers}\:\mathrm{of}\:\mathrm{types} \\ $$$$\mathrm{4k}+\mathrm{2}\:\&\:\mathrm{4k}+\mathrm{3}\:\mathrm{are}\:\mathrm{NOT}\:\mathrm{perfect}\:\Box\mathrm{s}. \\ $$ Answered by mr W last updated on 19/Dec/21 $$\left(\mathrm{1}\right)\:{type}\:\mathrm{4}{k}+\mathrm{3} \\ $$$${assume}\:{it}\:{can}\:{be}\:{a}\:{perfect}\:{square}.…

5-10-mod-11-

Question Number 95868 by rb222 last updated on 28/May/20 $$\mathrm{5}^{\mathrm{10}} \left({mod}\:\mathrm{11}\right)=? \\ $$ Commented by Rasheed.Sindhi last updated on 28/May/20 $$\mathrm{5}^{\mathrm{5}} \equiv\mathrm{1}\left({mod}\:\mathrm{11}\right) \\ $$$$\left(\mathrm{5}^{\mathrm{5}} \right)^{\mathrm{2}}…

Question-95697

Question Number 95697 by i jagooll last updated on 27/May/20 Answered by john santu last updated on 27/May/20 $${a}_{\mathrm{0}} =\:\frac{\mathrm{1}}{\mathrm{2}\pi}\:\underset{−\pi} {\overset{\pi} {\int}}\:\mathrm{cos}\:{at}\:{dt}\:=\:\frac{\mathrm{1}}{\mathrm{2}\pi{a}}\:\left[\mathrm{sin}\:{at}\:\right]_{−\pi} ^{\pi} \\ $$$$=\:\frac{\mathrm{sin}\:\pi{a}}{{a}\pi}\:.…