, 2010

Since $ab$ is divisible by $2c$ and $ca$ is divisible by $5b$, $ab \cdot ca$ must be divisible by $2c \cdot 5b$, hence $a^2$ is divisible by $2 \cdot 5$. Therefore $a^2$ is divisible by $2$ and $5$, hence $a$ is divisible by $2$ and $5$. Similarly $b$ is divisible by $2$ and $3$, and $c$ is divisible by $3$ and $5$. Consequently $abc$ is divisible by $2 \cdot 5 \cdot 2 \cdot 3 \cdot 3 \cdot 5 = 900$. On the other hand, $a = 10$, $b = 6$ and $c = 15$ satisfy the conditions and $abc = 900$.