Question

Cm: 2^51-1 chia hết cho 7

Answer 1

Dễ ợt đâu :))

 \(2^{51}-1=\left(2+2^2+2^3+.....+2^{51}\right)-\left(1+2+2^2+....+2^{50}\right)\)

Ta có :

\(2+2^2+2^3+....+2^{51}\)

\(=\left(2+2^2+2^3\right)+\left(2^4+2^5+2^6\right)+....+\left(2^{49}+2^{50}+2^{51}\right)\)

\(=2\left(1+2+2^2\right)+2^4\left(1+2+2^2\right)+....+2^{49}\left(1+2+2^2\right)\)

\(=2.7+2^4.7+....+2^{49}.7\)

\(=7\left(2+2^4+....+2^{49}\right)⋮7\)(1)

Chứng minh tương tự ta cũng có : \(\left(1+2+2^2+....+2^{50}\right)⋮7\)(2)

Từ (1) ; (2) \(\Rightarrow\left(2+2^2+2^3+.....+2^{51}\right)-\left(1+2+2^2+....+2^{50}\right)⋮7\)

Hay \(2^{51}-1⋮7\)(đpcm)