#### Question

If a,b are natural numbers such that 2a+ab+\dfrac{a}{b} =243

find the values of a,b

Question

If a,b are natural numbers such that 2a+ab+\dfrac{a}{b} =243

find the values of a,b

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