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D=1+.....+4^11chia het cho 5
D=(1+4)+(4^2+4^3)+......+(4^10+4^11)chia het cho 5
D=(1+4)+4^2(1+4)+....+4^10(1+4)chia het cho 5
D=5+4^2.5+....+4^10.5chia het cho 5
D=5(4^2+4^4+....+4^10)chia het cho 5
suy ra Dchia het cho 5 (do 5 chia het cho 5)
vậy Dchia het cho 5
a)A=(2+22)+(23+24)+...(29+210)
A=2(2+1)+23(1+2)+....+29(2+1)
A=3(2+23+25+27+29)
Vay A chia het cho 3(khi chia 3 duoc 2+23+25+27+29du 0)
b)A=(2+22+23+24+25)+(26+27+28+29+210)
A=2(1+2+22+23+24)+26(1+2+22+23+24)
A=31(2+26) luon chia het cho 31 :))
\(S=1+2+2^2+...+2^{99}\)
\(S=\left(1+2\right)+\left(2^2+2^3\right)+...+\left(2^{98}+2^{99}\right)\)
\(S=3+2^2.3+...+2^{98}.3\)
\(=3\left(1+2^2+...+2^{98}\right)⋮3\)
\(A=\left(2+2^2+2^3+2^4+2^5\right)+\)\(\left(2^6+2^7+2^8+2^9+2^{10}\right)+....\left(2^{86}+2^{87}+2^{88}+2^{89}+2^{90}\right)\)
\(A=2.\left(1+2+2^2+2^3+2^4\right)+2^6.\left(1+2+2^2+2^3+2^4\right)\)\(+....+2^{86}.\left(1+2+2^2+2^3+2^4\right)\)
\(A=2.21+2^6.21+...+2^{86}.21\)
\(A=21.\left(2+2^6+...+2^{86}\right)⋮21\)
\(A=1+3+3^2+.....+3^{11}\)
\(A=\left(1+3+3^2\right)+....+\left(3^9+3^{10}+3^{11}\right)\)
\(A=\left(3^0.1+3^0.3+3^0.3^2\right)+....+\left(3^9.1+3^9.3+3^9.3^2\right)\)
\(A=1.\left(1+3+3^2\right)+....+3^9\left(1+3+3^2\right)\)
\(A=1.13+....+3^9.13\)
\(A=13.\left(1+....+3^9\right)⋮13\left(đpcm\right)\)
a)
C=1+3+32+33+34+35+...+311
C=(1+3+32)+(33+34+35)+...+(39+310+311)
C=13+(33.1+33.3+33.32)+...+(39.1+39.3+39.32)
C=13+33.(1+3+32)+...+39.(1+3+32)
C=13.1+33.13+...+39.13
C=13.(1+33+35+37+39)\(⋮\)3
\(\Rightarrow\)C\(⋮\)3
Câu b ghép 4 số lại với nhau rồi làm như trên