a) \(C=\left(2+2^2\right)+\left(2^3+2^4\right)+...+\left(2^{59}+2^{60}\right)\)
\(=2\left(1+2\right)+2^3\left(1+2\right)+...+2^{59}\left(1+2\right)\)
\(\Rightarrow C⋮3\)
b) \(C=\left(2+2^2+2^3\right)+\left(2^4+2^5+2^6\right)+...+\left(2^{58}+2^{59}+2^{60}\right)\)
\(=2\left(1+2+4\right)+2^4\left(1+2+4\right)+...+2^{58}\left(1+2+4\right)\)
\(\Rightarrow C⋮7\)
c) \(C=\left(2+2^2+2^3+2^4\right)+\left(2^5+2^6+2^7+2^8\right)+...+\left(2^{57}+2^{58}+2^{59}+2^{60}\right)\)
\(=2\left(1+2+4+8\right)+2^5\left(1+2+4+8\right)+...+2^{57}\left(1+2+4+8\right)\)
\(\Rightarrow C⋮15\)
a) \(2+2^2+2^3+...+2^{60}\)
\(=\left(2+2^2\right)+\left(2^3+2^4\right)+...+\left(2^{59}+2^{60}\right)\)
\(=6+2^2\left(2+2^2\right)+...+2^{30}\left(2+2^2\right)\)
\(=2.3+2^2.2.3+2^4.2.3+...+2^{30}.2.3\)
\(=3\left(2+2^3+2^5+2^7+...+2^{31}\right)\)
\(\Rightarrow C⋮3\left(đpcm\right)\)
b) \(C=2+2^2+2^3+...+2^{60}\)
\(=\left(2+2^2+2^3\right)+\left(2^4+2^5+2^6\right)+...+\left(2^{58}+2^{59}+2^{60}\right)\)
\(=14+2^3\left(2+2^2+2^3\right)+...+2^{20}\left(2+2^2+2^3\right)\)
\(=2.7+2^3.2.7+2^6.2.7+...+2^{20}2.7\)
\(=7\left(2+2^4+...+2^{21}\right)\)
\(\Rightarrow C⋮7\left(đpcm\right)\)
c) \(C=2+2^2+2^3+...+2^{60}\)
\(=\left(2+2^2+2^3+2^4\right)+...+\left(2^{57}+2^{58}+2^{59}+2^{60}\right)\)
\(=30+2^4\left(2+2^2+2^3+2^4\right)+...+2^{15}\left(2+2^2+2^3+2^4\right)\)
\(=30+2^4.30+...+2^{15}.30\)
\(=30\left(2^4+2^8+...+2^{15}\right)\)
\(\Rightarrow C⋮15\left(đpcm\right)\)
a) C=(2+22)+(23+24)+...+(259+260)C=(2+22)+(23+24)+...+(259+260)
=2(1+2)+23(1+2)+...+259(1+2)=2(1+2)+23(1+2)+...+259(1+2)
⇒C⋮3⇒C⋮3
b) C=(2+22+23)+(24+25+26)+...+(258+259+260)C=(2+22+23)+(24+25+26)+...+(258+259+260)
=2(1+2+4)+24(1+2+4)+...+258(1+2+4)=2(1+2+4)+24(1+2+4)+...+258(1+2+4)
⇒C⋮7⇒C⋮7
c) C=(2+22+23+24)+(25+26+27+28)+...+(257+258+259+260)C=(2+22+23+24)+(25+26+27+28)+...+(257+258+259+260)
=2(1+2+4+8)+25(1+2+4+8)+...+257(1+2+4+8)=2(1+2+4+8)+25(1+2+4+8)+...+257(1+2+4+8)
⇒C⋮15⇒C⋮15
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