<s>[INST] $a, b$ are positive whole numbers such that $a + b = a/b + b/a$. How many possible values can $a^4 + b^4$ be? $a, b$ are positive whole numbers such that $a + b = a/b + b/a$. How many possible values can $a^4 + b^4$ be?
I tried using $(a^2 + b^2)^2$ but I don't know what to do after.
 [/INST] $a+b=\frac ab+\frac ba\iff a+b=\frac{a^2+b^2}{ab}\implies a^2b+ab^2=a^2+b^2\implies a^2\underbrace{(b-1)}_{\ge 0}+b^2\underbrace{(a-1)}_{\ge 0}=0$
Since the equation has $a,b$ both on denominator, we can reject zero as an admissible value for $a$ or $b$.
The only possibility for the sum above to be $0$ is that each term is zero, leading to $a=1$ and $b=1$.
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