13.437 What whole positive number should 180 be divided by, so that the remainder is $25\%$ of the quotient?
Solution. Let $n$ be the required number, $m$ be the remainder of the division of 180 by $n$, and $k$ be the quotient.

Then $\left\{\begin{array}{l}180=n \cdot k+m, \\ 4 m=k, \\ m4 m n>4 m^{2}$, i.e., $m^{2}<45 \Rightarrow m=4 \Rightarrow 4 n+1=45 \Rightarrow n=11$.

Answer: 11.