. For a positive integer $n$, let $S(n)$ denote the sum of its digits. Find the largest possible value of the expression $\frac{S(n)}{S(16 n)}$.

## Answer: 13
It is obvious that $S(a b) \leq S(a) S(b)$ for all positive integers $a$ and $b$. From here we get

$$
S(n)=S(n \cdot 10000)=S(16 n \cdot 625) \leq S(16 n) \cdot 13 \text {; }
$$

so we get $\frac{S(n)}{S(16 n)} \leq 13$.

For $n=625$ we have an equality. So the largest value is 13 .