\(S=1+2+2^2+2^3+...+2^{59}\)
\(S=\left(1+2\right)+\left(2^2+2^3\right)+...+\left(2^{58}+2^{59}\right)\)
\(S=3+2^2\cdot3+...+2^{58}\cdot3\)
\(S=3\cdot\left(1+2^2+...+2^{58}\right)\)
S chia hết cho 3
_____
\(S=1+2+2^2+...+2^{59}\)
\(S=\left(1+2+2^2\right)+\left(2^3+2^4+2^5\right)+...+\left(2^{57}+2^{58}+2^{59}\right)\)
\(S=7+7\cdot2^3+...+7\cdot2^{57}\)
\(S=7\cdot\left(1+2^3+...+2^{57}\right)\)
S chia hết cho 7
_____
\(S=1+2+2^2+2^3+...+2^{59}\)
\(S=\left(1+2+2^2+2^3\right)+\left(2^4+2^5+2^6+2^7\right)+...+\left(2^{56}+2^{57}+2^{58}+2^{59}\right)\)
\(S=15+2^4\cdot15+...+2^{56}\cdot15\)
\(S=15\cdot\left(1+2^4+...+2^{56}\right)\)
S chia hết cho 15
