We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.More informationAgree
ad
profileMalliKarjuna 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?
like 1 dislike 0 comment
profileRaj
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
$(4,4),(0,0),(3,6),(1,-2)$
like 0 dislike 0 comment

Answer This Question

?
Name
    cancel
    preview