smc

The right side of the equation can be rewritten as $111T=37\cdot 3T$. With trial and error and prime factorization as a guide, we can test different digits of $T$ to see if we can find two two-digit numbers that have the same units digit and multiply to $111T$. The only possibility that works is $37\cdot 27=999$. That means $E+M+T+Y=\boxed{21}$.