13.238. The shooter at the shooting range was offered the following conditions: each hit on the target is rewarded with five tokens, but for each miss, three tokens are taken away. The shooter was not very accurate. After the last ($n$-th) shot, he had no tokens left. How many shots were in the series and how many were successful, if $10<n<20$?
## Solution.

Let there be $k$ hits, and thus $n-k$ misses.

Then $5 k-3(n-k)=0,8 k=3 n \Rightarrow k=\frac{3 n}{8}$. Since $n$ and $k$ are natural numbers, $n=8 m$. For $10<n<20$, the only suitable value is $n=16$. Therefore, 16 shots were made, 6 of which were successful.

Answer: out of 16 shots, 6 were successful.