Beautiful little number theory prob

Question

Solving (a,b) + [a,b] = ab for natural a,b. How many possible a's are there.

I only know that (a,b)[a,b] = ab. Tried factoring out (a,b), but can't derive from it.

Answer 1

Let $(a,b)=g, a=pg,b=qg$ we have $(p,q)=1,[a,b]=pqg$. The equation becomes $g+pqg=pqg^2$. $$g+pqg=pqg^2$$ $$1+pq=pqg$$ $$pq(g-1)=1$$ When all these numbers are positive integers, the only way it can be done is $g=2,pq=1$. That means $a$ and $b$ are both 2.

Answer 2

First, you can take a look at some very simple cases. For instance, what happens if $a\neq b$.

We would then have a prime number $p$ dividing $b$ but not $a$ (or some prime power, in that case the argument works the same way).

Now, after we divide both sides by $GCD(a,b)$, you can ask the following questions:

Is the left side divisible by $p$? Why/not? How about the right side?

After finding out what that tells you about $p$ and hence $a,b$, you are nearly done.