B = 21 + 22 + 23 + ...+ 260
B = (21 + 22 + 23 + 24) + ( 25 + 26 + 27 + 28 ) + ... + ( 257 + 258 + 259 + 260)
B= 21 . ( 1+2+22+23) + 25. ( 1 + 2+22 + 23 ) + ... + 257 . ( 1+ 2+ 22 + 23 )
B= 21 . 30 + 25 . 30 + ... + 257 . 30
B = 30. ( 21 + 25 + ...+ 257 )
=> B chia hết cho 30
Cho B = 21 + 22+ 23 + .........+ 260
= ( \(^{2^1+2^2+2^3+2^4}\)) + .........+ \(^{2^{57}+2^{58}+2^{59}+2^{60}}\)
=
\(B=2^1+2^2+2^3+...+2^{60}\)
\(B=\left(2^1+2^2+2^3+2^4\right)+\left(2^5+2^6+2^7+2^8\right)+...+\left(2^{57}+2^{58}+2^{59}+2^{60}\right)\)
\(B=2^1\left(1+2+2^2+2^3\right)+2^5\left(1+2+2^2+2^3\right)+...+2^{57}\left(1+2+2^2+2^3\right)\)
\(B=2^1\left(1+2+4+8\right)+2^5\left(1+2+4+8\right)+...+2^{57}\left(1+2+4+8\right)\)
\(B=2^1\cdot15+2^5\cdot15+...+2^{57}\cdot15\)
\(B=30+2^4\cdot30+...+2^{56}\cdot30\)
\(B=30\left(1+2^4+...+2^{56}\right)\)
\(\Rightarrow B⋮30\)
Vậy \(B⋮30\left(đpcm\right)\)
