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![](https://rs.olm.vn/images/avt/0.png?1311)
a: \(A=\left(1+3\right)+...+3^{10}\left(1+3\right)\)
\(=4\left(1+...+3^{10}\right)⋮4\)
![](https://rs.olm.vn/images/avt/0.png?1311)
b: \(B=16^5+2^{15}\)
\(=\left(2^4\right)^5+2^{15}\)
\(=2^{20}+2^{15}\)
\(=2^{15}\left(2^5+1\right)=2^{15}\cdot33⋮33\)
c: \(45⋮9;99⋮9;180⋮9\)
Do đó: \(45+99+180⋮9\)
=>\(C⋮9\)
d: \(D=2+2^2+2^3+...+2^{60}\)
\(=\left(2+2^2+2^3\right)+\left(2^4+2^5+2^6\right)+...+\left(2^{58}+2^{59}+2^{60}\right)\)
\(=2\left(1+2+2^2\right)+2^4\cdot\left(1+2+2^2\right)+...+2^{58}\left(1+2+2^2\right)\)
\(=7\left(2+2^4+...+2^{58}\right)⋮7\)
\(D=2+2^2+2^3+...+2^{60}\)
\(=\left(2+2^2+2^3+2^4\right)+\left(2^5+2^6+2^7+2^8\right)+...+\left(2^{57}+2^{58}+2^{59}+2^{60}\right)\)
\(=2\left(1+2+2^2+2^3\right)+2^5\left(1+2+2^2+2^3\right)+...+2^{57}\left(1+2+2^2+2^3\right)\)
\(=15\left(2+2^5+...+2^{57}\right)\)
=>D chia hết cho cả 3 và 5
![](https://rs.olm.vn/images/avt/0.png?1311)
A = 8⁸ + 2²⁰
= (2³)⁸ + 2²⁰
= 2²⁴ + 2²⁰
= 2²⁰.(2⁴ + 1)
= 2²⁰.17 ⋮ 17
Vậy A ⋮ 17
![](https://rs.olm.vn/images/avt/0.png?1311)
Bài 1
a, cm : A = 165 + 215 ⋮ 3
A = 165 + 215
A = (24)5 + 215
A = 220 + 215
A = 215.(25 + 1)
A = 215. 33 ⋮ 3 (đpcm)
b,cm : B = 88 + 220 ⋮ 17
B = (23)8 + 220
B = 216 + 220
B = 216.(1 + 24)
B = 216. 17 ⋮ 17 (đpcm)
c, cm: C = 1 - 2 + 22 - 23 + 24 - 25 + 26 -...-22021 + 22022 : 6 dư 1
C=1+(-2+22-23+24- 25+26)+...+(-22017+22018-22019+22020-22021+22022)
C = 1 + 42 +...+ 22016.(-2 + 22 - 23 + 24 - 25 + 26)
C = 1 + 42+...+ 22016.42
C = 1 + 42.(20+...+22016)
42 ⋮ 6 ⇒ C = 1 + 42.(20+...+22016) : 6 dư 1 đpcm
![](https://rs.olm.vn/images/avt/0.png?1311)
a: \(G=8^8+2^{20}\)
\(=2^{24}+2^{20}\)
\(=2^{20}\left(2^4+1\right)=2^{20}\cdot17⋮17\)
b: Sửa đề: \(H=2+2^2+2^3+...+2^{60}\)
\(=2\left(1+2\right)+2^3\left(1+2\right)+...+2^{59}\left(1+2\right)\)
\(=3\left(2+2^3+...+2^{59}\right)⋮3\)
\(H=2+2^2+2^3+...+2^{60}\)
\(=2\left(1+2+2^2\right)+2^4\left(1+2+2^2\right)+...+2^{58}\left(1+2+2^2\right)\)
\(=7\left(2+2^4+...+2^{58}\right)⋮7\)
\(H=2+2^2+2^3+...+2^{60}\)
\(=\left(2+2^2+2^3+2^4\right)+...+\left(2^{57}+2^{58}+2^{59}+2^{60}\right)\)
\(=2\left(1+2+2^2+2^3\right)+...+2^{57}\left(1+2+2^2+2^3\right)\)
\(=15\left(2+2^5+...+2^{57}\right)⋮15\)
c: \(E=\left(1+3+3^2\right)+3^3\left(1+3+3^2\right)+...+3^{1989}\left(1+3+3^2\right)\)
\(=13\left(1+3^3+...+3^{1989}\right)⋮13\)
\(E=1+3+3^2+3^3+...+3^{1991}\)
\(=\left(1+3+3^2+3^3+3^4+3^5\right)+\left(3^6+3^7+3^8+3^9+3^{10}+3^{11}\right)+...+3^{1986}+3^{1987}+3^{1988}+3^{1989}+3^{1990}+3^{1991}\)
\(=364\left(1+3^6+...+3^{1986}\right)⋮14\)
![](https://rs.olm.vn/images/avt/0.png?1311)
a, Ta có 16 5 + 2 15 = 2 4 5 + 2 15 = 2 20 + 2 15 = 2 15 2 5 + 1 = 2 15 . 33 chia hết cho 33
b, Ta có: 8 8 + 4 10 = 2 3 8 + 2 2 10 = 2 24 + 2 20 = 2 20 2 4 + 1 = 2 20 . 17 chia hết cho 17
![](https://rs.olm.vn/images/avt/0.png?1311)
c) C = 5 + 52 + 53 +...+ 58
= ( 5 + 52 ) + ( 53 + 54 ) + ( 55 + 56 ) + ( 57 + 58 )
= 5 + 52 + 52( 5 + 52 ) + 54( 5 + 52 ) + 56( 5 + 52 )
= 5 + 52 ( 1 + 52 + 54 + 56 )
= 30. ( 1 + 52 + 54 + 56 ) chia hết cho 30
Vậy C = 5 + 52 + 53 +...+ 58 chia hết cho 30
b) B = 165 + 215
= (24)5 + 215
= 220 + 215
= 215. 25 + 215
= 215(25 + 1)
= 215.33 chia hết cho 33
Vậy B = 165 + 215 chia hết cho 33
a) A = 1 + 3 + 32 + .... + 311
= (1+3+32 ) + ( 33 + 34 + 35) + ..... + (39 + 310 + 311)
= 13 + 33 . 13 + .... + 39 . 13
= 13 . (1+ 33 +....+ 39)
=> A chia hết cho 13
b) B = 165 + 215
= 220 +215
= 215 . 25 + 215
= 215 . ( 25 + 1)
= 215 .33
=> B chia hết cho 33
c) C= 5 + 52 + 53 + .....+ 58
= (5 + 52) + (53 + 54) +....+ ( 57 + 58)
= 30 + 52 (5 + 52) + ....+ 56 ( 5 + 52)
= 30 + 52 . 30 + .....+ 56 . 30
= 30. ( 1+ 52 +....+ 56 )
=> C chia hết cho 30
d) D= 45 + 99+ 180 chia hết cho 9
Do 45 chia hết cho 9
99 chia hết cho 9
180 chia hết cho 9
=> 45 + 99 + 180 chia hết cho 9
e) E = 1+ 3 + 32 + 33 + ......+ 3199
= (1+3+32) + (33 + 34 + 35) +......+ (3197 + 3198 + 3199)
= 13 + 33 (1+3+32) +.......+ 3197(1+3+32)
= 13 + 33 . 13 + ..... + 3197 .13
= 13. ( 1+ 33 +....+ 3197)
=> E chia hết cho 13
f)
Ta có: 1028 + 8 = 100...08 (27 chữ số 0)
Xét 008 chia hết cho 8 => 1028 + 8 chia hết cho 8 (1)
Mà 1+27.0+ 8 = 9 chia hết cho 9 => 1028 + 8 chia hết cho 9 (2)
Mà (8,9) =1 (3)
Từ (1); (2); (3) => 1028 + 8 chia hết cho (8.9)= 72
g)
ta có: G= 88 + 220 = (23)8 + 220 = 224 + 220 = 220 . 24 + 220 = 220 . (24 + 1) = 220 . 17
=> G chia hết cho 17
a) A = 1 + 3 + 3^2 + ... + 3^11
A = ( 1 + 3 + 3^2 ) + ... + ( 3^9 + 3^10 + 3^11 )
A = 1(1 + 3 + 3^2 ) + ... + 3^9 ( 1 + 3 + 3^2 )
A = 1 . 13 + ... + 3^9 . 13
A = 13 ( 1 + ... + 3^9 ) chia hết cho 13
còn mấy ý kia bạn chỉ cần tách nhóm rồi làm tương tự là ok
Good luck