Question

fie S=1+3^1+3^2+.......3^203 Arata ca S divizibil cu 4 sau cu 13 sau cu 10

Answer 1

nr termeni 204
daca ii grupa cate 2
S=(1+3^1)+(3^2+3^3)+.......+(3^202+3^203 )=
=(1+3)+3^2(1+3)+.......+3^202(1+3)=
=4(1+3^2+...+3^202) deci divizibil 4


204:3=68  deci ii putem grupa cate 3
S=(1+3^1+3^2)+(3^3+3^4+3^5)+.......+(3^201+3^202+3^203 )=
=(1+3+9)+3^3(1+3+9)+.......+3^201(1+3+9)=
=13(1+3^3+.......+3^201) deci divizibil cu 13


204:4=51  deci ii putem grupa cate 4
S=(1+3^1+3^2+3^3)+.......+(3^200+3^201+3^202+3^203 )=
=(1+3+9+27)+.......+3^200(1+3+9+27)=
=40(1+3^4+.......+3^200)=
=4×10×(1+3^4+.......+3^200) deci divizibil cu 10