a, 6¹⁰⁰ - 1 = (6 . 6 . 6 ..... 6) - 1 = [(...6) . (...6) . (...6) ..... (...6)] - 1 = (...6) - 1 = ...5 \(⋮\) 5

b, 21²⁰ - 11¹⁰ = (21 . 21 . 21 . 21 . 21..... 21) - (11 . 11 . 11 . 11 ..... 11) = [(...1) . (...1) . (...1) . (...1).....(...1)] - [(...1) . (...1) . (...1) . (...1).....(...1)] = (...1) - (...1) = ....0 \(⋮\) 2; \(⋮\) 5

Ta có: \(A=2+2^2+2^3+\cdots+2^{120}\)

\(=\left(2+2^2+2^3\right)+\left(2^4+2^5+2^6\right)+\cdots+\left(2^{118}+2^{119}+2^{120}\right)\)

\(=2\left(1+2+2^2\right)+2^4\left(1+2+2^2\right)+\cdots+2^{118}\left(1+2+2^2\right)\)

\(=7\left(2+2^4+\cdots+2^{118}\right)\) ⋮7

Ta có: \(A=2+2^2+2^3+\cdots+2^{120}\)

\(=\left(2+2^2+2^3+2^4+2^5\right)+\left(2^6+2^7+2^8+2^9+2^{10}\right)+\cdots+\left(2^{116}+2^{117}+2^{118}+2^{119}+2^{120}\right)\)

\(=2\left(1+2+\cdots+2^4\right)+2^6\left(1+2+\cdots+2^4\right)+\cdots+2^{116}\left(1+2+\cdots+2^4\right)\)

\(=31\left(2+2^6+\ldots+2^{116}\right)\) ⋮31

Ta có: \(A=2+2^2+2^3+\cdots+2^{120}\)

\(=\left(2+2^2\right)+\left(2^3+2^4\right)+\cdots+\left(2^{119}+2^{120}\right)\)

\(=2\left(1+2\right)+2^3\left(1+2\right)+\cdots+2^{119}\left(1+2\right)=3\left(2+2^3+\cdots+2^{119}\right)\) ⋮3

Ta có: A⋮3

A⋮7

mà ƯCLN(3;7)=1

nên A⋮3*7

=>A⋮21

Ta có: \(A=2+2^2+2^3+...+2^{120}\)

\(=\left(2+2^2+2^3\right)+\left(2^4+2^5+2^6\right)+...+\left(2^{118}+2^{119}+2^{120}\right)\)

\(=14+2^3\cdot14+...+2^{117}\cdot14\)

\(=14\cdot\left(1+2^3+...+2^{117}\right)⋮7\)

Ta có: \(A=2+2^2+2^3+...+2^{120}\)

\(=\left(2+2^2+2^3+2^4+2^5\right)+\left(2^6+2^7+2^8+2^9+2^{10}\right)+...+\left(2^{116}+2^{117}+2^{118}+2^{119}+2^{120}\right)\)

\(=62+2^5\cdot62+...+2^{115}\cdot62\)

\(=62\cdot\left(1+2^5+...+2^{115}\right)⋮31\)

Ta có: \(A=2+2^2+2^3+...+2^{120}\)

\(=\left(2+2^2+2^3+2^4+2^5+2^6\right)+\left(2^7+2^8+2^9+2^{10}+2^{11}+2^{12}\right)+...+\left(2^{115}+2^{116}+2^{117}+2^{118}+2^{119}+2^{120}\right)\)

\(=126+126\cdot2^6+...+126\cdot2^{114}\)

\(=126\cdot\left(1+2^6+...+2^{114}\right)⋮21\)

\(10^{2120}=1000\ldots0\) (2120 chữ số 0)

Tổng các chữ số của số \(10^{2120}\) là:

1+0+0+...+0=1

=>\(10^{2120}\) chia 3 dư 1(1)

2120=2118+2=3*706+3

=>2120 chia 3 dư 2(2)

Từ (1),(2) suy ra \(10^{2120}+2120\) ⋮3

Ta có: \(10^{2120}\) ⋮10

2120⋮10

Do đó: \(10^{2120}+2120\) ⋮10

mà \(10^{2120}+2120\) ⋮3

và ƯCLN(3;10)=1

nên \(10^{2120}+2120\) ⋮3*10

=>\(10^{2120}+2120\) ⋮30

\(1,8^8+2^{20}=2^{24}+2^{20}=2^{20}\left(2^4+1\right)=2^{20}\cdot17⋮17\)

\(2,A=\left(2+2^2\right)+\left(2^3+2^4\right)+...+\left(2^{119}+2^{120}\right)\\ A=2\left(1+2\right)+2^3\left(1+2\right)+...+2^{119}\left(1+2\right)\\ A=3\left(2+2^3+...+2^{119}\right)⋮3\)

\(A=\left(2+2^2+2^3\right)+...+\left(2^{118}+2^{119}+2^{120}\right)\\ A=2\left(1+2+2^2\right)+...+2^{118}\left(1+2+2^2\right)\\ A=\left(1+2+2^2\right)\left(2+...+2^{118}\right)=7\left(2+...+2^{118}\right)⋮7\\ A=\left(2+2^2+2^3+2^4\right)+...+\left(2^{117}+2^{118}+2^{119}+2^{120}\right)\\ A=2\left(1+2+2^2+2^3\right)+...+2^{117}\left(1+2+2^2+2^3\right)\\ A=\left(1+2+2^2+2^3\right)\left(2+...+2^{117}\right)=15\left(2+...+2^{117}\right)⋮15\)

Mọi người giải giúp em với ạ. Em đang cần gấp !!!

Ta có; 1110 = (10+1)10 ( khai triển nhị thức Niu- tơn )

Do đó 1110 -1 chia hết cho 100

\(A=\left(2+2^2+2^3\right)+...+\left(2^{118}+2^{119}+2^{120}\right)\\ A=2\left(1+2^2+2^3\right)+...+2^{118}\left(1+2^2+2^3\right)\\ A=\left(1+2^2+2^3\right)\left(2+...+2^{118}\right)\\ A=7\left(2+...+2^{118}\right)⋮7\)

\(A=2\left(1+2+2^2\right)+2^4\left(1+2+2^2\right)+...+2^{118}\left(1+2+2^2\right)\)

\(=2.7+2^4.7+...+2^{118}.7=7\left(2+2^4+...+2^{118}\right)⋮7\)