If $$a,b$$ are natural numbers such that $$2a+ab+\dfrac{a}{b} =243$$
find the values of $$a,b$$

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If $$a,b$$ are natural numbers such that $$2a+ab+\dfrac{a}{b}  =243$$
find the values of $$a,b$$

Uniteasy Admin · Accepted Answer

Let
$$k =\dfrac{a}{b}$$ , 
then $$a=bk$$,
$$2a+ab+\dfrac{a}{b} $$
$$ = 2bk+kb^2+k$$
$$=k(b+1)^2$$
On the other hand,
$$243 = 3\cdot 9^2 =27\cdot 3^2 $$
Then
$$k = 3$$,$$b = 8, a = 24$$
or
$$k = 27$$,$$b = 2, a = 54$$
Therefore
there are two value sets for the expression, $$b = 8, a = 24$$ or $$b = 2, a = 54$$

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