STARS wrote: Find all pairs of positive integers $ (a, b)$ such that \[ ab = gcd(a, b) + lcm(a, b). \] $ (a;b)=k$----->$ a=nk;b=mk$-----$ (m,n)=1$---->$ [a;b]=nmk$----->$ ab = gcd(a, b) + lcm(a, b)=(a;b)+[a;b]$-----> $ nmk^{2}=k+mnk$-----> $ mnk=1+mn$----->$ k=\frac{1}{mn}+1$---->$ mn=1$-----> $ m=n=1$ and $ k=2$----> $ a=b=2$

It is well known that $ ab = \gcd(a,b)lcm(a,b)$, so $ \gcd(a,b)lcm(a,b) = \gcd(a,b) + lcm(a,b)$ $ (\gcd(a,b) - 1)(lcm(a,b) - 1) = 1$ $ \gcd(a,b) = lcm(a,b) = 2$. $ (a,b) = (2,2)$