\(5x^2+2xy+y^2-16x+16=0\)

=>\(x^2+2xy+y^2+4x^2-16x+16=0\)

=>\(\left(x+y\right)^2+\left(2x-4\right)^2=0\)

=>\(\left\{{}\begin{matrix}x+y=0\\2x-4=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=2\\y=-2\end{matrix}\right.\)

\(x\) = 2; \(y\) = 2; \(z\) = 5.

\(11^y\equiv1\left(mod10\right)\Rightarrow11^y+29⋮10\)

\(\Rightarrow272x⋮10\Rightarrow272x⋮5\)

\(\Rightarrow x⋮5\Rightarrow x=5\) do x nguyên tố

Thay vào phương trình:

\(272.5=11^y+29\Rightarrow11^y=1331\Rightarrow y=3\)

Vậy \(\left(x;y\right)=\left(5;3\right)\)

Toán cô Hương BG ấy gì thảo nào quen quen

\(\Leftrightarrow\sqrt{9x^2+16x+96}=3x-16y-24\)

Vế phải nguyên \(\Rightarrow\) vế trái nguyên

\(\Rightarrow9x^2+16x+96=k^2\)

\(\Rightarrow81x^2+144x+864=\left(3k\right)^2\)

\(\Leftrightarrow\left(9x+8\right)^2+800=\left(3k\right)^2\)

\(\Leftrightarrow\left(3k-9x-8\right)\left(3k+9x+8\right)=800\)

Pt ước số thật kinh dị với số ước của 800 

Ta có \(9x^2+16x+96=\left(3x-24-16y\right)^2\)

\(\Leftrightarrow9x^2+16x+96=9x^2-6x\left(16y+24\right)+\left(16y+24\right)^2\)\(\Leftrightarrow16x+96=\left(16y+24\right)\left(16y+24-6x\right)\)

\(\Leftrightarrow8\left(2x+12\right)=4\left(4y+6\right).2\left(8y+12-3x\right)\)

\(\Leftrightarrow2x+12=\left(4y+6\right)\left(8y+12-3x\right)\)\(\Leftrightarrow2x+12=32y^2+48y-12xy+48y+72-18x\)

\(\Leftrightarrow32y^2+96y-12xy-20x+60=0\)\(\Leftrightarrow32y^2+96y+60=12xy+20x\)\(\Leftrightarrow8y^2+24y+15=3xy+5x\)

\(\Leftrightarrow8y^2+24y+15=x\left(3y+5\right)\)\(\Leftrightarrow x=\dfrac{8y^2+24y+15}{3y+5}\)

\(\Leftrightarrow9x=\dfrac{9\left(8y^2+24y+15\right)}{3y+5}=\dfrac{72y^2+216y+135}{3y+5}\)\(=\dfrac{\left(72y^2+120y\right)+\left(96y+160\right)-25}{3y+5}\)\(=24y+32-\dfrac{25}{3y+5}\)

\(\Leftrightarrow24y+32-\dfrac{25}{3y+5}\in Z\)\(\Rightarrow3y+5\in U\left(25\right)=\left\{\pm1,\pm5,\pm25\right\}\)\(\Leftrightarrow3y\in\left\{-4,-6,-10,0,-30,20\right\}\)\(\Rightarrow y\in\left\{-2,-10,0\right\}\)

+) Với y=-2=> x=1

+) với y=-10=> x=-23    

Vậy pt cho 2 cặp (x,y) nguyên =(1,-2),(-23,-10)

KO TỀM ĐC VÌ X NGUYÊN TỐ THÌ Y KO NGUYÊN TỐ .(CHƯA CHẮC ĐÃ DÚNG NHA)