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\(M=3^1+3^2+3^3+...+3^{28}+3^{29}+3^{30}\)
\(M=3\left(1+3+3^2\right)+3^4\left(1+3+3^2\right)...+3^{28}\left(1+3+3^2\right)\)
\(M=3.13+3^4.13...+3^{28}.13\)
\(M=13.\left(3+3^4...+3^{28}\right)⋮13\)
\(\Rightarrow dpcm\)
\(A=\left(1+2+2^2\right)+\left(2^3+2^4+2^5\right)+...+\left(2^{27}+2^{28}+2^{29}\right)\\ A=\left(1+2+2^2\right)+2^3\left(1+2+2^2\right)+...+2^{27}\left(1+2+2^2\right)\\ A=\left(1+2+2^2\right)\left(1+2^3+...+2^{27}\right)\\ A=7\left(1+2^3+...+2^{27}\right)⋮7\)
\(S=2^1+2^2+2^3+2^4+2^5+2^6+..+2^{28}+2^{29}+2^{30}\)
\(S=2.\left(1+2+2^2\right)+2^4.\left(1+2+2^2\right)+...+2^{28}.\left(1+2+2^2\right)\)
\(S=\left(1+2+2^2\right).\left(2+2^4+...+2^{28}\right)\)
\(S=7.\left(2+2^4+...+2^{28}\right)\)
⇒ \(S⋮7\) ( điều phải chứng minh )
\(a,A=5^1+5^2+...+5^{100}\)
\(\Rightarrow A=5\left(1+5\right)+5^3\left(1+5\right)+...+5^{99}\left(1+5\right)\)
\(\Rightarrow6\left(5+5^3+...+5^{99}\right)\)
\(\Rightarrow A⋮6\)
\(b,B=2+2^2+2^3+...+2^{28}+2^{29}+2^{30}\)
\(\Rightarrow B=2\left(1+2+2^2\right)+...+2^{28}\left(1+2+2^2\right)\)
\(\Rightarrow7\left(2+...+2^{28}\right)\)
\(\Rightarrow B⋮7\)
a) (1+5+52+53+...529)chia hết cho 6
Đặt (1+5+52+53+...529) = A
\(A=\left(1+5\right)+\left(5^2+5^3\right)+\left(5^4+5^5\right)....+\left(5^{28}+5^{29}\right)\)
\(A=\left(1+5\right)+5^2\left(5+1\right)+5^4\left(5+1\right)+...+5^{28}\left(5+1\right)\)
\(A=6+5^2.6+5^4.6+...+5^{28}.6\)
Vậy A chia hết cho 6
b) (1+3+3^2+3^3+...+3^29) chia hết cho 13
Đặt B= (1+3+3^2+3^3+...+3^29)
\(B=\left(1+3+3^2\right)+\left(3^3+3^4+3^5\right)+...+\left(3^{27}+3^{28}+3^{29}\right)\)
\(B=13+3^3\left(1+3+3^2\right)+....+3^{27}\left(1+3+3^2\right)\)
\(B=13+3^3.13+....+3^{27}.13\)
Vậy B chia hết 13
Câu c,d tương tự.Chúc bạn học tốt
Ta có \(M=\left(3^1+3^2+3^3\right)+\left(3^4+3^5+3^6\right)+...+\left(3^{28}+3^{29}+3^{30}\right)\)
\(=3\left(1+3+3^2\right)+3^4.\left(1+3+3^2\right)+...+3^{28}.\left(1+3+3^2\right)\)
\(=13\left(3+3^4+...+3^{28}\right)⋮13\Rightarrow M⋮13\)
M = 31 + 32 + 33 +...+ 328 + 329 + 330
M = ( 31 + 32 + 33) + ...+ ( 328 + 329 + 330 )
M = 3(1 + 3 + 32 ) +...+ 328( 1 + 3 + 32)
M = 3 .13 +...+ 328.13
\(\Rightarrow M⋮13\)(đpcm)
!!!
1 + 2 + 23 + 24 +...+ 228 + 229
= 20 + 21 + 23 + 24 +...+ 228 + 229
= (20 + 21) + (23 + 24) +...+ (228 + 229)
= 20(20 + 21) + 23(20 + 21) +...+ 228(20 + 21)
= 20 . 3 + 23 . 3 +...+ 228 . 3
= (20 + 23 + 26 +...+ 228) . 3 chia hết cho 3
1 + 2 + 22 + 23 +...+ 228 + 229
= 20 + 21 + 22 + 23 +...+ 228 + 229
= (20 + 21) + (22 + 23) +...+ (228 + 229)
= 20(20 + 21) + 22(20 + 21) +...+ 228(20 + 21)
= 20 . 3 + 22 . 3 +...+ 228 . 3
= (20 + 22 + 24 +...+ 228) . 3 chia hết cho 3
Hi hi. Mình nhầm tí.