Question

[Thiên bình cute]

Tìm các số nguyên dương x, y thỏa mãn x³ -xy=y³ +25

Answer 1

[ミ★Zero ❄ ( Hoàng Nhật )]

\(x^3-xy=y^3+25\)

Ta có :

\(x^3-y^3=xy+25\)

\(\Leftrightarrow\left(x-y\right)^3+3xy\left(x-y\right)=xy+25\)

Đặt \(x-y=a;\)\(xy=b\)Ta có :

\(a^3+3ab=b+25\)

\(\Leftrightarrow a^3-25=-b\left(3a-1\right)\)

\(\Leftrightarrow27\left(a^3-25\right)⋮3a-1\)

\(\Leftrightarrow27a^3-1-674⋮3a-1\)

Do \(27a^3-1⋮3a-1\Rightarrow674⋮3a-1\)

mà \(674=2.337\)

Nên \(3a-1\in\left\{\pm1;\pm2;\pm337;\pm674\right\}\)

Do \(3a-1⋮3\)( Dư 2 ) 

Nên \(3a-1\in\left\{-1;2;-337;674\right\}\)

\(\Rightarrow a\in\left\{0;1;-112;225\right\}\)

Ta có : \(b=a^3-25̸\)\(1-3a\)

\(\left(a,b\right)=\left(0,-25\right);\left(1,12\right);\left(-112;4169\right);\left(225;-16900\right)\)

Vì \(\left(x-y\right)^2+4xy\ge0\Rightarrow a^2+4b\ge0\)Vì vậy chỉ có a = 1 , y = 12 . \(\Rightarrow x-y=1;xy=12\)

Vậy \(\left(x,y\right)=\left(4,3\right);\left(-3,-4\right)\)