My birthday Puzzle, part 2

$\overline{uvw} \times \overline{SAY} = \overline{HAPPY} + \overline{BIRTH} +\overline{DAY}$

$\overline{uvw}$ is a 3-digit number. Each letter (other than $u, v, w$ ) represents a different digit from 0 to 9.

Suppose there is a unique solution to this puzzle. How many such 3-digit $\overline{uvw}$ ?

Example : It is known that if $\overline{uvw}=508$ , there are two solutions to the puzzle. Hence 508 is not the 3-digit number that we interested.

You may try My birthday Puzzle, part 1 .