If $m$ and $n$ are positive integers such that $\gcd(m,n) = 12$, then what is the smallest possible value of $\gcd(10m,15n)$?
Since $\gcd(m,n) = 12$, both $m$ and $n$ are divisible by 12.  Then $10m$ is divisible by $10 \cdot 12 = 120$, and $15n$ is divisible by $12 \cdot 15 = 180$.  Since 60 divides both 120 and 180, $\gcd(10m,15n)$ must be at least 60.

If we set $m = n = 12$, then $\gcd(m,n) = \gcd(12,12) = 12$, and $\gcd(10m,15n) = \gcd(120,180) = 60$, which shows that the value of 60 is attainable.  Therefore, the smallest possible value of $\gcd(10m,15n)$ is $\boxed{60}$.