jmc

We know that $\gcd (a,b)\cdot \mathrm{lcm}[a,b]=ab$ for all positive integers $a$ and $b$. Hence, in this case, $ab=180$. The prime factorization of 180 is $2^2\cdot 3^2\cdot 5$, so $a=2^p\cdot 3^q\cdot 5^r$ and $b=2^s\cdot 3^t\cdot 5^u$ for some nonnegative integers $p$, $q$, $r$, $s$, $t$, and $u$. Then $ab=2^{p+s}\cdot 3^{q+t}\cdot 5^{r+u}$. But $ab=180=2^2\cdot 3^2\cdot 5$, so $p+s=2$, $q+t=2$, and $r+u=1$.

We know that $\gcd (a,b)=2^{\min \{p,s\}}\cdot 3^{\min \{q,t\}}\cdot 5^{\min \{r,u\}}$. The possible pairs $(p,s)$ are $(0,2)$, $(1,1)$, and $(2,0)$, so the possible values of $\min \{p,s\}$ are 0 and 1. The possible pairs $(q,t)$ are $(0,2)$, $(1,1)$, and $(2,0)$, so the possible values of $\min \{q,t\}$ are 0 and 1. The possible pairs $(r,u)$ are $(0,1)$ and $(1,0)$, so the only possible value of $\min \{r,u\}$ is 0.

Therefore, the possible values of $\gcd (a,b)$ are $2^0\cdot 3^0=1$, $2^1\cdot 3^0=2$, $2^0\cdot 3^1=3$, and $2^1\cdot 3^1=6$, for a total of $\boxed{4}$ possible values.