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showing 1-1 of 1 (answers : 1, comments : 0),   sorted newest to the oldest
Question
How many pairs of integers are there such that twice the sum of the integers is equal to their product?
 
(1) (0) Comment Answer
permutations & combinationsnumbers
2016-10-15 15:39:57 
MalliKarjuna Reddy
1 Answer
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
$(4,4),(0,0),(3,6),(1,-2)$
 
(0) (0) Comment
2016-10-16 06:51:39 
Raj (Senior Maths Expert, careerbless.com)
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