Question

Dau coroana, masinute, tractorase , inimioare , etc .Arătați că 2^1 + 2^2 + …+ 2^120 se divide cu 15.

Answer 1

Fie S=2^1+2^2+2^3+2^4...+2^120

Grupez termenii câte 3, dând factor p3 fel mai mic

S=(2^1+2^2+2^3+2^4)+...+(2^117+2^118+2^119+2^120)

S=2^1(1+2+2^2+2^3)+...+2^117(1+2+2^2+2^3)

S=(1+2+2^2+2^3)(2^1+2^5+...+2^117)

S=(1+2+4+8)(2^1+2^5+...+2^117)

S=15*(2^1+2^5+...+2^117) |15