Question

Positive integers a and b are such that a+b=a/b +b/a . What is the value of $a^2+b^2?$

Answer 1

$a+b=a/b+b/a⟹a+b=\frac{a^2+b^2}{ab}$

$a^{2}b+a{b}^2=a^2+b^2$

$a^2(b-1)+b^2(a-1)=0$

$\because a,b$ are positive integers $,a^2,b^2$ can't be zero

$⟹a-1=0⟹a=1,b-1=0⟹b=1$

$a^2+b^2=1+1=2$