Question

Chứng minh rằng

B = 2 + 2² + 2³ + 2⁴ + ... + 2¹⁰⁰ chia hết cho 31

C = 53! - 51! chia hết cho 29

Answer 1

+) ta có : \(B=2+2^2+2^3+2^4+...+2^{100}\)

\(\Leftrightarrow B=\left(2+2^2+...+2^5\right)+\left(2^6+2^7+...+2^{10}\right)...+\left(2^{96}+2^{97}+2^{98}+2^{99}+2^{100}\right)\)

\(\Leftrightarrow B=2\left(1+2+...+2^4\right)+2^6\left(1+2+...+2^4\right)+...+2^{96}\left(1+2+...+2^4\right)\)

\(\Leftrightarrow B=2\left(31\right)+2^6\left(31\right)+...+2^{96}\left(31\right)=31\left(2+2^6+...+2^{96}\right)⋮31\left(đpcm\right)\)

+) ta có : \(C=53!-51!=53.52.51!-51!=51!\left(53.52-1\right)\)

\(\Leftrightarrow C=51!\left(2755\right)=29.95.51!⋮29\left(đpcm\right)\)