For how many ordered pairs of positive integers \((m,n)\), does \(m \cdot n\) divide \(2008 \cdot 2009 \cdot 2010\)? Express your answer as a product of powers of prime numbers in the form \(2^k \cdot 3^m \cdot 5^n\). Find the value of \(k + m + n\).

Let's think step by step and output the final answer within \boxed{}.
user