$279$
31. (5 points)
Tiger Multiplier uses 3 different digits $a, b, c$ to form 6 two-digit numbers $\overline{a b}, \overline{a c}, \overline{b a}, \overline{b c}, \overline{c a}, \overline{c b}$, and he finds that the sum of these 6 two-digit numbers is exactly equal to $(a+2) \times(b+2) \times(c+2)$. What is the smallest three-digit number $\overline{a b c}$?