KVPY SA Exam 2016 Question Paper

n = ab
$\mathit{a}\mathit{b}={\mathit{a}}^2+{\mathit{b}}^3$
$\Rightarrow 10\mathit{a}+\mathit{b}={\mathit{a}}^2+{\mathit{b}}^3$
⇒ a (10 - a) + b (1 - b) (1 + b) = 0
⇒ a (10 – a) = (b – 1) (b) (b + 1)
If b = 2; a (10 – a) = 6 ⇒ no value of 'a'
b = 3; a (10 – a) = 24 ⇒ a ∈ {4, 6}.
{nos. are 43 & 63}
b = 4; a (10 – a) = 60 ⇒ no value of a
b = 5; a (10 – a) = 120 ⇒ no need to check further
∴ nos. are 43 & 63.