A=\(2^0+2^1+2^2+2^3+2^4+...+2^{2016}\)

\(=\left(2^0+2^1+2^2\right)+\left(2^3+2^4+2^5+......+2^{2016}\right)\)

=7+\(2^3\left(1+2^1+2^2+....+2^{2012}+2^{2013}\right)\)

\(=7+8\left(1+2+...+2^{2013}\right)\)

Vì \(8\left(1+2+...+2^{2012}+2^{2013}\right)⋮8\)

Mà \(7⋮̸8\) Nên \(7+8\left(1+2+..+2^{2012}+2^{2013}\right)\div8\) sẽ dư 7

Hay \(A\div8\) dư 7

a) 52(-43)-52(55)-104

=52(-43)-52*55-52*2

=52(-43-55-2)

=52(-100)=-5200

b) \(x^2-3xy+x-3y=7\)

\(\Leftrightarrow x\left(x-3y\right)+\left(x-3y\right)=7\)

\(\Leftrightarrow\left(x-3y\right)\left(x+1\right)=7\)

Mà 7={1*7;7.1;-1*-7;-7*-1}

Do đó ta lập bảng

P=(a-1)(b-1)-(ab+2)

=ab-a-b+1-ab+2

=-a-b+3

Ta có (-98)(1-246)-246*98

=98*(246-1)-246*98

=98(246-1-246)

=98*(-1)=-98

Bài 2: chia 10n cho 5n-3 như bình thường ta được dư là 6

Để A có giá trị nguyên thì \(10n⋮5n-3\) Do đó 6 phai chia hết cho 3n+2

<= >5n-3\(\in u\left(6\right)=\left\{\pm1;\pm2;\pm3;\pm6\right\}\\\)

Lập bảng

a) \(\left(-8\right)\cdot\left(-3\right)^3\cdot125\)

=\(\left(-2\right)^3\cdot\left(-3\right)^3\cdot5^3\)

=\(\left(-2\cdot-3\cdot5\right)^3\)

=\(30^3\)

b) \(27\cdot\left(-2\right)^3\cdot\left(-7\right)\cdot49\)

=\(3^3\cdot\left(-2\right)^3\cdot\left(-7\right)\cdot7^2\)

=\(3^3\cdot\left(-2\right)^3\cdot\left(-7\right)^3\)

\(=\left(3\cdot-2\cdot-7\right)^3=42^3\)

ta có \(57^{1999}=57^{1996}\cdot57^3=\left(57^4\right)^{499}\cdot57^3=\overline{.....01}^{499}\cdot\overline{...93}=\overline{...3}\)

Vậy \(57^{1999}\) có tận cùng là 3

A=\(2^0+2^1+....+2^{100}\)

=>2A=\(2^1+2^2+2^3+.....+2^{100}+2^{101}\)

=>2A-A=\(\left(2^1+2^2+....+2^{100}+2^{101}\right)-\left(2^0+2^1+....+2^{100}\right)\)

\(\Leftrightarrow A=2^{101}-1\)

Ta có \(2^{20}=4^{10}\equiv76\) (mod 100) =>\(\left(2^{20}\right)^5=2^{100}\equiv76\) (mod 100)

=>\(2^{100}\cdot2\)\(\equiv76\cdot2=152\) (mod 100)

Hay \(2^{101}\equiv152\) (mod 100)

=>\(2^{101}-1\equiv152-1=151\) (mod 100)

Lại có 151 chia cho 100 dư 51

Do đó A chia cho 100 dữ51

-38(25-4)+25(-4+38)

=-38*25+38*4-25*4+25*38

=[-38*25+25*38]+[38*4-25*4]

=38*4-25*4=(38-25)*4=13*4=52