Greatest Common Divisor of two numbers

If $ab=600$ how large can the greatest common divisor of $a$ and $b$ be?

I am not sure if I should check for all factor multiples of $a$ and $b$ for this question. Please advise.

We have $600=2^3\cdot 5^2\cdot 3$. To make the gcd large, we give a $2$ to each of $a$ and $b$, also a $5$. So the largest possible gcd is $10$.
Remark: The idea generalizes. Let $n$ have prime power factorization $$n=p_1^{d_1}p_2^{d_2}\cdots p_k^{d_k}.$$ Let $e_i=\lfloor d_i/2\rfloor$, where $\lfloor x\rfloor$ is the greatest integer that is $\le x$. Then the greatest possible gcd of $a$ and $b$, where $ab=n$, is $$p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}.$$
• You are welcome. For the maximum, we distribute the power of any prime $p$ as evenly as possible between $a$ and $b$. – André Nicolas Jun 10 '13 at 2:00
$$ab=600=2^3\cdot 3\cdot 5^2$$ The greatest common divisor of $a,b$ will have the greatest number of common prime factors that you can arrange.