Răspuns:
a)
[tex]10^{30}=(10^{3})^{10}=1000^{10}[/tex]
[tex]2^{100}=(2^{10})^{10}=1024^{10}[/tex]
[tex]2^{100}>10^{30}[/tex]
[tex]10^{28}=(10^{4})^{7}=10000^{7}[/tex]
[tex]2^{91}=(2^{13})^{7}=8192^{7}[/tex]
[tex]10^{28}>2^{91}[/tex]
b)
[tex]2^{100}=2^{91}*2^{9}=8192^{7}*512[/tex]
[tex]10^{31}=10^{28}*10^{3}=10000^{7}*1000[/tex]
[tex]10^{30}<2^{100}<10^{31}[/tex]
[tex]10^{30}[/tex] are 31 cifre
[tex]2^{100}[/tex] are 31 cifre
[tex]10^{31}[/tex] are 32 cifre (este cel mai mic numar de 32 cifre)
c)
Aflam ultimele trei cifre ale nr [tex]2^{100}[/tex]
[tex]2^{100}=(2^{10})^{10}=1024^{10}=1048576^{5}[/tex]
...576*...576=...776
...776*...576=...976
...976*...576=...176
...476*...576=...376
Ultimele 3 cifre ale nr [tex]2^{100}[/tex] sunt 376
Daca are in total 31 de cifre suma maxima posibila este 999...999376=28*9+16=268
Sau calculam ultimele doua cifre care sunt ...76 si mai stim ca prima cifra nu poate sa fie 9 ci maxim 8.
Deci suma maxima posibila este
899..9976=8+28*9+13=273
