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.


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