Solution: $35+1=36=2^{2} \cdot 3^{2}$. By Theorem 5.15, the number of 35-complete partitions with the smallest number of parts is $\frac{(2+2)!}{2!2!}=6$, and their part numbers are
$$
2(2-1)+2(3-1)=6 .
$$
Example 5.7 How many 35-complete partitions with the smallest number of parts are there? What are their part numbers?