$16.2 .21^{\star}$ A certain positive integer's square ends with two identical non-zero digits. Find the smallest positive integer with this property.
Let $p=10 a \pm b(a, b \in \mathbf{Z}, 1 \leqslant b \leqslant 5)$, then $p^{2} \equiv 10 \cdot( \pm 2 a b)+$ $b^{2}(\bmod 100)$.

It is not difficult to verify that only when $b=2$, the last two digits of $p^{2}$ have the same parity. The required $p$ must be in the form of $10 a \pm 2$.

It is easy to verify that among $12,18,22,28,32,38, \cdots$, the smallest positive integer whose square has the same last three digits is $38\left(38^{2}=1444\right)$.