Find all integers $x,y$ which satisfy the equation $xy=20-3x+y$.
Problem
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Tags: number theory, Diophantine equation
28.04.2021 18:23
Firstly, we have $\displaystyle xy-y=y( x-1) =20-3x$ Then we just have to solve this division equation $\displaystyle y=\frac{20-3x}{x-1} \in \mathbb{Z}$
28.04.2021 18:30
xy=20-3x+y xy+3x-y-3=17 x(y+3)-1(y+3)=17 (y+3)(x-1)=17=can be 17*1 or 1*17 or (-1)*(-17) or (-17)*(-1) from this we get the soln as (y,x)=(14,2),(-2,18),(-20,0),(-4,-16).
28.04.2021 18:32
NCEE wrote: Attachment which application/website is this?
28.04.2021 18:33
geometrylover123 wrote: Firstly, we have $\displaystyle xy-y=y( x-1) =20-3x$ Then we just have to solve this division equation $\displaystyle y=\frac{20-3x}{x-1} \in \mathbb{Z}$ Continuation: $y=\frac{20-3x}{x-1}=\frac{3-3x+17}{x-1}=-1+\frac{17}{x-1}\in \mathbb{Z}\Rightarrow x-1\in \left\{ \pm 1,\pm 17 \right\}$
28.04.2021 18:47
@SK1729 AoPS User Desmos.
@jasperE3 AoPS User This is a Diophantine equation, perfect for SFFT.
01.06.2021 10:38
Okay. Lamboreghini wrote:
@SK1729 AoPS User Desmos.
@jasperE3 AoPS User This is a Diophantine equation, perfect for SFFT.
01.06.2021 19:59