jmc

The condition $r_9(5n)\le 4$ can also be stated as $‘‘5n\equiv 0,1,2,3,\text{ or }4(\mathrm{mod}9)."$'

We can then restate that condition again by multiplying both sides by $2:$ $$10n\equiv 0,2,4,6,\text{ or }8(\mathrm{mod}9).$$This step is reversible (since $2$ has an inverse modulo $9$). Thus, it neither creates nor removes solutions. Moreover, the left side reduces to $n$ modulo $9,$ giving us the precise solution set $$n\equiv 0,2,4,6,\text{ or }8(\mathrm{mod}9).$$We wish to determine the $22^{\text{nd}}$ nonnegative integer in this solution set. The first few solutions follow this pattern: $$\begin{array}{ccccc}0& 2& 4& 6& 8\\ 9& 11& 13& 15& 17\\ 18& 20& 22& 24& 26\\ 27& 29& 31& 33& 35\\ 36& 38& \cdots & & \end{array}$$The $22^{\text{nd}}$ solution is $\boxed{38}.$