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Category: Set Theory

1-3-5-7-9-2005-mod-1000-

Question Number 203490 by cortano12 last updated on 20/Jan/24 $$\:\:\:\:\mathrm{1}×\mathrm{3}×\mathrm{5}×\mathrm{7}×\mathrm{9}×…×\mathrm{2005}\:=\:…\:\left(\mathrm{mod}\:\mathrm{1000}\right) \\ $$ Answered by AST last updated on 20/Jan/24 $${x}=\mathrm{1}×\mathrm{3}×\mathrm{5}…×\mathrm{2005}\equiv\mathrm{0}\left({mod}\:\mathrm{125}\right) \\ $$$${x}\equiv\left(\mathrm{1}×\mathrm{3}×\mathrm{5}×\mathrm{7}\right)^{\mathrm{250}} ×\mathrm{1}×\mathrm{3}×\mathrm{5}\left({mod}\:\mathrm{8}\right)\equiv\mathrm{7}\left({mod}\:\mathrm{8}\right) \\ $$$${x}=\mathrm{125}{q}\equiv\mathrm{7}\left({mod}\:\mathrm{8}\right)\Rightarrow\mathrm{5}{q}\equiv\mathrm{15}\left({mod}\:\mathrm{8}\right)\Rightarrow{q}\equiv\mathrm{3}\left({mod}\:\mathrm{8}\right)…

Question-199624

Question Number 199624 by pascal889 last updated on 06/Nov/23 Commented by mr W last updated on 06/Nov/23 $${you}\:{have}\:{posted}\:{the}\:{same}\:{question} \\ $$$${one}\:{time}\:{and}\:{a}\:{comment}\:{was}\:{given}. \\ $$$${why}\:{don}'{t}\:{you}\:{check}\:{if}\:{your}\:{question} \\ $$$${is}\:{wrong}?\:{maybe}\:{the}\:{question}\:{is} \\…

Prove-that-pi-2-0-ln-1-sint-sint-dt-pi-2-8-1-2-arccos-2-

Question Number 196950 by Erico last updated on 05/Sep/23 $$\mathrm{Prove}\:\mathrm{that}\:\underset{\:\mathrm{0}} {\int}^{\:\frac{\pi}{\mathrm{2}}} \frac{\mathrm{ln}\left(\mathrm{1}+\alpha\mathrm{sin}{t}\right)}{\mathrm{sin}{t}}{dt}=\:\frac{\pi^{\mathrm{2}} }{\mathrm{8}}−\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{arccos}\alpha\right)^{\mathrm{2}} \\ $$ Answered by Mathspace last updated on 06/Sep/23 $${f}\left({x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{ln}\left(\mathrm{1}+{xsint}\right)}{{sint}}{dt}…