Question

1.Determinati nr nat abc( cu bara de asupra),stiind ca impartite la 7 dau catul bc(cu bara de asupra)si restul a.
Comparati nr:2^222+3^111 si 2^111+3^222

Answer 1

1. $\overline{abc} :7=\overline{bc}~~rest ~~a$
$\overline{abc}=7*\overline{bc}+a$
100a+10b+c=70b+7c+a
99a-60b-6c=0
33a-20b-2c=0
=> a =par si a<8
=> pt a=2=> b=3 si c=3
 pt a=4=> b=6 si c=6
 pt a=6=> b=9 si c=9
=> $\overline{abc}$∈{233,466,699}

2.
2²²²+3¹¹¹ =(2¹¹¹)²+3¹¹¹
2¹¹¹+3²²²=2¹¹¹+(3¹¹¹)²
=> 2²²²+3¹¹¹<2¹¹¹+3²²²