[tex]9)~\text{Comparati:} \\ \\
a)~2^{28} ~cu~ 5^{14} \\
2^{28} = 2^{2\times 14} = (2^2)^{14}= 4^{14} \\
4^{14} \ \textless \ 5^{14} \\
\Longrightarrow~~2^{28} \ \textgreater \ 5^{14} \\ \\
b) ~3^{51} cu 4^{34} \\
3^{51} = 3^{3\times 17} = 27^{17} \\
4^{34} = 4^{2\times 17} = 16^{17} \\
27^{17} \ \textgreater \ 16^{17} \\
\Longrightarrow~~3^{51} \ \textgreater \ 4^{34} \\ \\
c)~2^{105}~cu~3^{63} \\
2^{105} = 2^{5 \times 21} = 32^{21} \\
3^{63}= 3^{3 \times 21} = 27^{21} \\
32^{21} \ \textgreater \ 27^{21} \\
\Longrightarrow~~2^{105}\ \textgreater \ 3^{63}
[/tex]
[tex]d)~4^{111}~cu~6^{74} \\
4^{111} = 4^{3 \times 37} = 64^{37} \\
6^74 = 6{2 \times 37} = 36^37 \\
64^{37} \ \textgreater \ 36^37 \\
\Longrightarrow~~4^{111}\ \textgreater \ 6^{74}[/tex]
[tex]11) \\
a)~ S=2^{14}+2^{16}+2^{18} = 2^14(1 + 2^2+2^4)=2^{14}\times 21 \\
\Longrightarrow~~S~\vdots ~ 2^{14} ~si~S~\vdots ~ 3~si~S~\vdots ~ 7 \\ \\
b) ~S=3^{15}+3^{16}+3^{17} = 3^{15}(1 + 3 + 3^2) =3^{15} \times 13 \\
\Longrightarrow~~S~\vdots ~ 3^{15} ~si~S~\vdots ~ 13
[/tex]
[tex]c)~S = 5^1+5^2+5^3+\hdots+5^{101}+5^{102} \\
Sirul are 102 termeni. \\
\text{Putem grupa sirul in grupe de cate 2 termeni }\\
\text{deoarece 102 este numar par.} \\
\text{Prima grupa este: } \\
5^1+5^2 = 5+25 = 30\\ \text{Celelalte grupe sunt multiplii ai primei grupe.} \\
\Longrightarrow~~S~\vdots ~ 30 \\
\Longrightarrow~~S~\vdots ~ 2~si~ S~\vdots ~ 3~si~ S~\vdots ~ 5
[/tex]
[tex]\text{Putem grupa sirul in grupe de cate 3 termeni }\\
\text{deoarece 102 este multiplu de 3.} \\
\text{Prima grupa este: } \\
5^1+5^2+5^3 = 5+25 +125= 155 \\
\text{Celelalte grupe sunt multiplii ai primei grupe.} \\
\Longrightarrow~~S~\vdots ~ 155 \\
\Longrightarrow~~S~\vdots ~ 5~si~ S~\vdots ~ 31
[/tex]
