\(P=2+2^2+2^3+...+2^{60}\\ =\left(2+2^2+2^3+2^4+2^5+2^6\right)+\left(2^7+2^8+2^9+2^{10}+2^{11}+2^{12}\right)+...+\left(2^{55}+2^{56}+2^{57}+2^{58}+2^{59}+2^{60}\right)\\ =1\left(2+2^2+2^3+2^4+2^5+2^6\right)+2^6\left(2+2^2+2^3+2^4+2^5+2^6\right)+...+2^{54}\left(2+2^2+2^3+2^4+2^5+2^6\right)\\ =\left(2+2^2+2^3+2^4+2^5+2^6\right)\left(1+2^6+...+2^{54}\right)\\ =63\left(1+2^6+...+2^{54}\right)\\ =3\cdot7\cdot3\left(1+2^6+...+2^{54}\right)\text{ chia hết cho 3 và 7}\)\(P=2+2^2+2^3+...+2^{60}\\ 2P=2\left(2+2^2+2^3+...+2^{60}\right)=2^2+2^3+2^4+...+2^{61}\\ 2P-P=\left(2^2+2^3+2^4+...+2^{61}\right)-\left(2+2^2+2^3+...+2^{60}\right)\\ P=2^{61}-2\\ P=2\left(2^{60}-1\right)\)
