a) \(\left(5n+7\right)\left(4n+6\right)\)

\(=\left(5n+7\right)4n+\left(5n+7\right)6\)

\(=20n^2+28n+30n+32\)

\(=20n^2+58n+32\)

Vì \(20n^2⋮2\) ; \(58n⋮2\) ; \(32⋮2\) nên \(\left(5n+7\right)\left(4n+6\right)⋮2\)

b) \(\left(8n+1\right)\left(6n+5\right)\)

\(=\left(8n+1\right)6n+\left(8n+1\right)5\)

\(=48n^2+6n+40n+5\)

\(=48n^2+46n+5\)

Vì \(\left(48n^2+46n\right)⋮2\) mà \(5⋮̸2\) nên \(\left(8n+1\right)\left(6n+5\right)⋮̸2\)

c) \(n\left(n+1\right)\left(2n+1\right)\)

\(=n\left(n+1\right)\left(n-1+n-2\right)\)

\(=n\left(n-1\right)\left(n+1\right)+n\left(n+1\right)\left(n+2\right)\)

Với \(\forall n\in N\), tích 3 số tự nhiên liên tiếp chia hết cho 6 nên \(n\left(n-1\right)\left(n+1\right)⋮6\) và \(n\left(n+1\right)\left(n+2\right)⋮6\)

Vậy \(n\left(n+1\right)\left(2n+1\right)⋮6\)

a, Ta có 8n - 59 = ( 2n -16 ) + ( 2n -16 ) + ( 2n - 16 ) + ( 2n - 16 ) + 5

2n - 16 luôn luôn chia hết cho 2n - 16 

=> 4.(2n-16) chia hết cho 2n-16 <=> 5 chia hết cho 2n - 16

=> 2n - 16 thuộc Ư(5) = { 1;-1;5;-5 }

Tự làm nốt

b, tương tự 

c, 6n - 46 = (2n-18) + (2n-18) + (2n-18) + 8

... Tiếp tục :))

a ,\(8n-59⋮2n-16\)

Mà \(2n-16⋮2n-16\) 

\(\Rightarrow4\left(2n-16\right)⋮2n-16\)

\(\Rightarrow8n-64⋮2n-16\) 

\(\Rightarrow\left(8n-59\right)-\left(8n-64\right)⋮2n-16\) 

\(\Rightarrow8n-59-8n+64⋮2n-16\) 

\(\Rightarrow5⋮2n-16\) 

\(\Rightarrow2n-16\inƯ\left(5\right)\) 

\(\Rightarrow2n-16\in\left\{\pm1;\pm5\right\}\) 

\(\Rightarrow2n\in\left\{17;15;21;11\right\}\) 

\(\Rightarrow\) KHÔNG CÓ SỐ NÀO THỎA MÃN CỦA 2n 

\(\Rightarrow x\in\varnothing\)

\(a,2n+3⋮6n+4\Leftrightarrow6n+9⋮6n+4\Leftrightarrow6n+9-6n-4⋮6n-4\Leftrightarrow5⋮6n-4\Leftrightarrow6n-4\in\left\{-5;5;1;-1\right\}\Leftrightarrow6n\in\left\{-1;9;5;-3\right\}\Leftrightarrow n\in\left\{-\dfrac{1}{6};1,5;\dfrac{5}{6};-0,5\right\}\)

a) \(\Rightarrow\left(6n+5\right)-2\left(3n-1\right)⋮3n-1\)

\(\Rightarrow\left(6n+5\right)-\left(6n-2\right)⋮3n-1\)

\(\Rightarrow6n+5-6n+2⋮3n-1\)

\(\Rightarrow7⋮3n-1\)

\(\Rightarrow3n-1\inƯ\left(7\right)=\left(1;-1;7;-7\right)\)

ta có bảng sau :

3n-1         1                        -1                              7                                -7

n              L                        0                               L                                -2

mà \(n\in Z\)

\(\Rightarrow n\in\left(0;-2\right)\)

b) \(\Rightarrow\left(2n-1\right)-2\left(n+1\right)⋮n+1\)

\(\Rightarrow\left(2n-1\right)-\left(2n+2\right)⋮n+1\)

\(\Rightarrow2n-1-2n-2⋮n+1\)

\(\Rightarrow-1⋮n+1\)

\(\Rightarrow n+1\inƯ\left(-1\right)=\left(1;-1\right)\)

ta có bảng sau

n+1                       1                                    -1

n                           0                                    -2

mà \(n\in Z\)

KL :\(n\in\left(0;-2\right)\)

c) \(\Rightarrow\left(9n-1\right)+9\left(9-n\right)⋮9-n\)

\(\Rightarrow\left(9n-1\right)+\left(81-9n\right)⋮9-n\)

\(\Rightarrow9n-1=81-9n⋮9-n\)

\(\Rightarrow80⋮9-n\)

\(\Rightarrow9-n\inƯ\left(80\right)=\left(1;-1;2;-2;4;-4;8;-8;10;-10;5;-5;20;-20;40;-40;80;-80\right)\)

ta có bảng sau :

9 - n        1      -1         2       -2       4       -4      5      -5    8     -8     10       -10      20       -20      40      -40       80       -80

n             8     10         7        11      5      13     4      14    1     17      -1        19     -11        29      -31      49       -71       89

Mà \(n\in Z\)

\(\Rightarrow n\in\left(8;10;7;11;5;13;4;14;1;17;-1;19;-11;29;-31;49;-71;89\right)\)

a/

\(x+6y⋮17\Rightarrow5\left(x+6y\right)=5x+30y⋮17\)

\(5x+47y=\left(5x+30y\right)+17y\)

\(5x+30y⋮17\left(cmt\right);17y⋮17\Rightarrow5x+47y⋮17\)

b/

\(3x+16y⋮5\Rightarrow2\left(3x+16y\right)=6x+32y=\left(5x+30y\right)+\left(x+2y\right)⋮5\)

Mà \(5x+30y⋮5\Rightarrow x+2y⋮5\)