Since we would like the product to have 303 digits, then we would like this product to be greater than $10^{302}$ but less than $10^{303}$.

We will start by trying $k=300$. In this case, $\left(2^{k}\right)\left(5^{300}\right)=\left(2^{300}\right)\left(5^{300}\right)=10^{300}$, so we want $k$ to be bigger than 300 .

Each time we increase $k$ by 1 , the existing product is multiplied by 2 . For the final product to have 303 digits, we need to multiply $10^{300}$ by a power of 2 between 100 and 1000 . The smallest power of 2 that satisfies this is $2^{7}=128$.

Therefore, we would like $k=307$. In this case,

$$
\left(2^{307}\right)\left(5^{300}\right)=\left(2^{7}\right)\left(2^{300}\right)\left(5^{300}\right)=\left(2^{7}\right)\left(10^{300}\right)=128 \times 10^{300}
$$

When this number is expanded, the digits are 128 followed by 300 zeros. Therefore, the sum of the digits is 11 .
If $k$ is the smallest positive integer such that $\left(2^{k}\right)\left(5^{300}\right)$ has 303 digits when expanded, then the sum of the digits of the expanded number is
(A) 11
(B) 10
(C) 8
(D) 7
(E) 5