2 Factorization Method
Example 2 Let $a, b, c$ be distinct natural numbers, and $a b^{2} c^{3}=1350$. Then the maximum value of $a+b+c$ is $\qquad$
(1990, Wu Yang Cup Junior High School Mathematics Competition)
Solution: $\because 1350=2 \times 5^{2} \times 3^{3}=150 \times 3^{2} \times 1^{3}$, $\therefore$ when $a=150, b=3, c=1$, the maximum value of $a+b+c$ is 154.