Greatest Common Divisor of natural numbers

Question

Suppose two natural numbers a, b satisfy ab = n for some fixed integer n. What is the maximum possible value of gcd(a, b)?

Let $d= gcd (a,b)$
So, $d= xa + yb$
I don't know how to proceed with this. Is the answer n?

Answer 1

If $d$ divides both $a$ and $b$, then $d^2$ divides $n$.

So the largest possible value for $d$ is the square root of the largest square factor of $n$.