Question

aratati ca numarul a=3^{81}+3^{82}=3^{84} nu este patrat perfect

Answer 1

 [tex]a=3^{81}\cdot(1+3+3^3)=3^{81}\cdot31 \;nu\; este\; p.p.\; \\ pentru\; ca\; u(3^{81})=3 \; \;si\; \;u(31)=1\;\\ \rightarrow\;\;u(3*1)=3\;p.p.\; nu \;se \;termina\; in \;3 !!! [/tex]

Answer 2

[tex]a= 3^{81} +3^{82}+3^{84} [/tex]
$3^{81}~are ~U.C.~3~pentru~ca~81:4~are~rest~1$
$3^{82}~are ~U.C.~9~pentru~ca~82:4~are~rest~2$
$3^{84}~are ~U.C.~1~pentru~ca~84:4~are~rest~0$
$=\ \textgreater \ a ~are ~U.C.~3~pt.~ca~3+9+1=13$
$Nici~un~patrat~perfect~nu~are~ultima~cifra~3$
$=\ \textgreater \ numarul~a~nu~este~patrat~perfect$