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a/ Ta co: \(B=3+3^3+3^5+...+3^{1987}+3^{1989}+3^{1991}\)
\(\Rightarrow B=\left(3+3^3+3^5\right)+...+\left(3^{1987}+3^{1989}+3^{1991}\right)\)
\(\Rightarrow B=3\cdot\left(1+3^2+3^4\right)+...+3^{1987}\cdot\left(1+3^2+3^4\right)\)
\(\Rightarrow B=3\cdot91+...+3^{1987}\cdot91\)
\(\Rightarrow B=91\cdot\left(3+...+3^{1987}\right)\)
\(\Rightarrow13\cdot7\cdot\left(3+...+3^{1987}\right)⋮13\left(dpcm\right)\)
\(M=2+2^3+2^5+2^7+....+2^{51}\)
\(=\left(2+2^3\right)+\left(2^5+2^7\right)+....+\left(2^{49}+2^{51}\right)\)
\(=10+2^4\left(2+2^3\right)+....+2^{48}\left(2+2^3\right)\)
\(=10+2^4.10+...+2^{48}.10\)
\(=10\left(1+2^4+...+2^{48}\right)\Rightarrow M⋮10\)
\(=2.5.\left(1+2^4+...+2^{48}\right)\Rightarrow M⋮5\)
\(M=2+2^3+2^5+2^7+....+2^{51}.\)
\(M+2^{ }=2+2+2^3+2^5+2^7+.....+2^{51}\)
\(=\left(2+2+2^3\right)+\left(2^5+2^7+2^9\right)+....+\left(2^{47}+2^{49}+2^{51}\right)\)
\(=12+2^4\left(2+2^3+2^5\right)+......+2^{46}\left(2+2^3+2^5\right)\)
\(=12+2^4.42+....+2^{46}.42\)
\(=12+7.3.2\left(2^4+...+2^{46}\right)\)
\(\Rightarrow M=\left[12+7.3.2\left(2^4+.....+2^{46}\right)\right]-2\)
\(=10+7.3.2\left(2^4+....+2^{46}\right)\)
Ta có: \(7.3.2\left(2^4+...+2^{46}\right)⋮7\)mà 10 không chia hết cho 7
Suy M không chia hết cho 7
30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 310 + 311
= ( 30 + 31 + 32 + 33 ) + ( 34 + 35 + 36 + 37 ) + ( 38 + 39 + 310 + 311 )
= ( 1 + 3 + 9 + 27 ) + 34 ( 1 + 3 + 9 + 27 ) + 38 ( 1 + 3 + 9 + 27 )
= 40 + 34.40 + 38.40
=40( 1 + 34 + 38 ) chia hết cho 40