Question

A = 2 + 2²+ 2³+ ... + 2¹⁰⁰chia hết cho 3 và 6

Answer 1

\(A=2+2^2+2^3+2^4+...+2^{100}\)

\(=\left(2+2^2\right)+\left(2^3+2^4\right)+...+\left(2^{99}+2^{100}\right)\)

\(=\left(2+2^2\right)+2^2\left(2+2^2\right)+...+2^{98}\cdot\left(2+2^2\right)\)

\(=6\left(1+2^2+...+2^{98}\right)⋮6\)

=>\(A=3\cdot2\cdot\left(1+2^2+...+2^{98}\right)⋮3\)