3. (1990 Japan Mathematical Olympiad Team Selection Test) There is a positive integer whose square has the last three digits as non-zero identical digits. Find the smallest positive integer with this property.

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3. 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 can be verified that when $b=1,3,4,5$, the tens and units digits of $p^{2}$ have opposite parity; only when $b=2$, the last two digits of $p^{2}$ have the same parity. Therefore, the required $P$ must be of the form $10 a \pm 2$, and for $p=12, p^{2}=144$, the last two digits are 4.
By calculating $12^{2}, 18^{2}, 22^{2}, 28^{2}, 32^{2}, 38^{2}$, we find that the smallest positive integer with the last three non-zero digits being the same is 38.