Alphanumeric Puzzle multplication : $OW\times HE=WWW$

Question

$OW\times HE=WWW$ where $O,W,H,E$ are digits.

I've gathered that

$$W\in\{2,3,4,5,6,7,8,9\}$$ $$E\in\{1,3,6,7,9\}$$

but I can't seem to move past it.

Answer 1

$WWW$ is a multiple of $111=3×37$, so $OW$ or $HE$ has to be a multiple of $37$. Thus four cases:

$HE=37\implies7×W\equiv W\bmod10\implies 6×W\equiv0\bmod10\implies W=5\implies15×37=555$

$HE=74\implies4×W\equiv W\bmod10\implies 3×W\equiv0\bmod10\implies$ no solution, we can't have $W=0$

$OW=74\implies74×HE=444\implies$ no solution, must have $HE\ge10$ to avoid a nonzero initial digit

$OW=37\implies$ let's see if the OP can work out the second solution.