Question

1)Aratati ca 2005^2005+2006^2006-1 este divizibil cu 10( ^ inseamna la puterea)
2)Aratati ca 7^n+7^n+1+7^n+2+7^n+3 este divizibil cu 10 (^ inseamna la puterea)

Answer 1

[tex]1)U.C.( 2005^{2005})=5;\longrightarrow~5~ridicat~la~orice~putere~da~un~numar~cu~ult.~cifra:~\boxed{5} \\~~U.C.(2006^{2006})=6;\longrightarrow~6~ridicat~la~orice~putere~da~un~numar~cu~ult.~cifra:~\boxed{6} \\ Deci,~u.c.~a~nr.=5+6-1=0\Rightarrow Numarul~este~divizibil~cu~10.\\
2)7^{n}+7^{n+1} +7^{n+2} +7^{n+3}=7^{n}\cdot (1+7+7^{2}+7^{3})=7^{n}\cdot400=7^{n}\cdot40\cdot\underline{10}~deci,~este~divizbil~cu~10.\\
\underline{Observatii}:1)~U.C.~inseamna~"ultima~cifra".\\
~~~~~~~~~~~~~~~~2)Acolo~am~dat~factor~comun~pe~7^{n}.[/tex]