Question

Cho A=2 + 2 mũ 2 + 2 mũ 3 + ... + 2 mũ 60,chưngs minh A chia hết cho 42

Answer 1

\(A=2+2^2+2^3+......+2^{60}\)

\(A=2^1+2^2+2^3+.......+2^{60}\)

\(A=\left(2^{60}-2^1\right):\left(2^2\right)\)

\(A=2^{58}\)

Answer 2

\(A=2+2^2+2^3+...+2^{60}\)

\(=\left(2+2^2\right)+\left(2^3+2^4\right)+...+\left(2^{59}+2^{60}\right)\)

\(=2\left(1+2\right)+2^3\left(1+2\right)+...+2^{59}\left(1+2\right)\)

\(=3\left(2+2^3+...+2^{59}\right)⋮3\).

\(A=2+2^2+2^3+...+2^{60}\)

\(=\left(2+2^2+2^3\right)+\left(2^4+2^5+2^6\right)+...+\left(2^{58}+2^{59}+2^{60}\right)\)

\(=\left(2+2^2+2^3\right)+2^3\left(2+2^2+2^3\right)+...+2^{57}\left(2+2^2+2^3\right)\)

\(=14\left(1+2^3+...+2^{57}\right)⋮14\)

Ta thấy \(\left(3,14\right)=1\)nên \(A\)chia hết cho \(3.14=42\).