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a) Ta có: \(A=3+3^3+3^5+...+3^{1991}\)
\(=\left(3+3^3+3^5\right)+\left(3^7+3^9+3^{11}\right)+...+\left(3^{1987}+3^{1989}+3^{1991}\right)\)
\(=3\times\left(1+3^2+3^4\right)+3^7\times\left(1+3^2+3^4\right)+...+3^{1987}\times\left(1+3^2+3^4\right)\)
\(=3\times91+3^7\times91+...+3^{1987}\times91\)
\(=3\times7\times13+3^7\times7\times13+...+3^{1987}\times7\times13\)
\(=13\times\left(3\times7+3^7\times7+...+3^{1987}\times7\right)\)
Vì \(A=13\times\left(3\times7+3^7\times7+...+3^{1987}\times7\right)\)nên A chia hết cho 13.
b) Ta có: \(A=3+3^3+3^5+...+3^{1991}\)
\(=\left(3+3^3+3^5+3^7\right)+...+\left(3^{1985}+3^{1987}+3^{1989}+3^{1991}\right)\)
\(=3\times\left(1+3^2+3^4+3^6\right)+...+3^{1985}\times\left(1+3^2+3^4+3^6\right)\)
\(=3\times820+...+3^{1985}\times820\)
\(=3\times20\times41+...+3^{1985}\times20\times41\)
\(=41\times\left(3\times20+...+3^{1985}\times20\right)\)
Vì \(A=41\times\left(3\times20+...+3^{1985}\times20\right)\)nên A chia hết cho 41.
B= 3+3^3+3^5+...+3^1991
a)Các số hạng của B là: (1991-1):2+1=996(số hạng)
b)
B=3+3^3+3^5+...+3^1991
B=(3+3^3+3^5)+(3^6+3^7+3^8)+...+(3^1989+3^1990+3^1991)
= 3(3^2+3^4+1)+3^6(3+3^2+1)+...+3^1989(3+3^2+1)
=3.91+3^6.13+...+3^1989.13
Ta thấy : 3.91 chia hết cho 91 => chia hết cho 13
3^6.13 chia hết cho 13.
....
3^1989.13 chia hết cho 13.
=> =3.91+3^6.13+...+3^1989.13 chia hết cho 13.
=> ĐPCM
\(A=2+2^2+......+2^{59}+2^{60}\)
\(A=2\left(1+2\right)+....+2^{59}\left(1+2\right)\)
\(A=2\cdot3+...+2^{59}\cdot3⋮3\)
\(2+2^2+2^3+....+2^{58}+2^{59}+2^{60}\)
\(=2\left(1+2+4\right)+....+2^{58}\left(1+2+4\right)\)
\(=2\cdot7+.....+2^{58}\cdot7⋮7\)
a) \(2010^{100}+2010^{99}\)
\(=2010^{99}\left(2010+1\right)\)
\(=2010^{99}.2011⋮2011\left(dpcm\right)\)
b) \(3^{1994}+3^{1993}-3^{1992}\)
\(=3^{1992}\left(3^2+3-1\right)\)
\(=3^{1992}.11⋮11\left(dpcm\right)\)
c) \(4^{13}+32^5-8^8\)
\(=\left(2^2\right)^{13}+\left(2^5\right)^5-\left(2^3\right)^8\)
\(=2^{26}+2^{25}-2^{24}\)
\(=2^{24}\left(2^2+2-1\right)\)
\(=2^{24}.5⋮5\left(dpcm\right)\)
a.
\(\left(0,25\right)^3\times32\)
\(=\left(0,25\right)^3\times2^5\)
\(=\left(0,25\right)^3\times2^3\times2^2\)
\(=\left(0,25\times2\right)^3\times4\)
\(=\left(0,5\right)^3\times4\)
\(=0,125\times4\)
\(=0,5\)
b.
\(\left(-0,125\right)^3\times80^4\)
\(=\left(-0,125\right)^3\times80^3\times80\)
\(=\left(-0,125\times80\right)^3\times80\)
\(=\left(-10\right)^3\times80\)
\(=-1000\times80\)
\(=-80000\)
c.
\(3^{1994}+3^{1993}-3^{1992}\)
\(=3^{1992}\times\left(3^2+3-1\right)\)
\(=3^{1992}\times\left(9+3-1\right)\)
\(=3^{1992}\times11\)
\(\Rightarrow3^{1994}+3^{1993}-3^{1992}⋮11\)
d.
\(4^{13}+32^5-8^8\)
\(=\left(2^2\right)^{13}+\left(2^5\right)^5-\left(2^3\right)^8\)
\(=2^{26}+2^{25}-2^{24}\)
\(=2^{24}\times\left(2^2+2-1\right)\)
\(=2^{24}\times\left(4+2-1\right)\)
\(=2^{24}\times5\)
\(\Rightarrow4^{13}+32^5-8^8⋮5\)
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