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MalliKarjuna Reddy

asked 2016-10-15 15:39:57

How many pairs of integers are there such that twice the sum of the integers is equal to their product?

Raj

answered 2016-10-16 06:51:39

Ans: $4$

Let the integers be $x$ and $y$

$2(x+y)=xy\\

\Rightarrow xy-2(x+y)=0~~\cdots(1)$

We know that, $(x-2)(y-2)=xy-2(x+y)+4~~\cdots(2)$

From $(1)$

$xy-2(x+y)+4=4\\

\Rightarrow (x-2)(y-2)=4$

Thus, only the following are the possibilities.

$(x-2)=2,~(y-2)=2\\

\implies x=4,~y=4$

$(x-2)=-2,~(y-2)=-2\\

\implies x=0,~y=0$

$(x-2)=1,~(y-2)=4\\

\implies x=3,~y=6$

$(x-2)=-1,~(y-2)=-4\\

\implies x=1,~y=-2$

i.e., only $4$ pairs of integers satisfies the given condition. These are

$(4,4),(0,0),(3,6),(1,-2)$

Let the integers be $x$ and $y$

$2(x+y)=xy\\

\Rightarrow xy-2(x+y)=0~~\cdots(1)$

We know that, $(x-2)(y-2)=xy-2(x+y)+4~~\cdots(2)$

From $(1)$

$xy-2(x+y)+4=4\\

\Rightarrow (x-2)(y-2)=4$

Thus, only the following are the possibilities.

$(x-2)=2,~(y-2)=2\\

\implies x=4,~y=4$

$(x-2)=-2,~(y-2)=-2\\

\implies x=0,~y=0$

$(x-2)=1,~(y-2)=4\\

\implies x=3,~y=6$

$(x-2)=-1,~(y-2)=-4\\

\implies x=1,~y=-2$

i.e., only $4$ pairs of integers satisfies the given condition. These are

$(4,4),(0,0),(3,6),(1,-2)$

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