Question

Aratati ca numarul a=2plus2²plus2³plus..........plus2⁶⁰ este divizibil cu 42 .

Answer 1

$\it a=2+2^2+2^3+\ ...\ +2^{60}\\ \\ 2+2^2+2^3+2^4+2^5+2^6=2+4+8+16+32+64=126=3\cdot42\\ \\ a=(2+2^2+2^3+2^4+2^5+2^6)+2^6(2+2^2+2^3+2^4+2^5+2^6)+\ ...\ +\\ \\ +2^{54}(2+2^2+2^3+2^4+2^5+2^6)=126\cdot(1+2^6+2^{12}+\ ...\ +2^{54})=\\ \\ =3\cdot42\cdot(1+2^6+2^{12}+\ ...\ +2^{54})\in M_{42} \Rightarrow a\ \vdots\ 42.$