A number theory problem by Lakshya Singh

Number Theory Level 3

Positive integers $a$ and $b$ satisfy the equation $a + b = \dfrac ab + \dfrac ba$ . What is the value of $a^2 + b^2$ .

1 solution

Brian Charlesworth
Sep 3, 2016

The given equation can be written as $ab(a + b) = a^{2} + b^{2} \Longrightarrow a^{2}(b - 1) = b^{2}(1 - a)$ .

Now as $a,b$ are positive integers we see that $a^{2}(b - 1) \ge 0$ and $b^{2}(1 - a) \le 0$ , so we require that

$a^{2}(b - 1) = b^{2}(1 - a) = 0 \Longrightarrow a = b = 1 \Longrightarrow a^{2} + b^{2} = \boxed{2}$ .