# Does this alphametic have only one solution?

I could only get one answer for the following alphametic. Can you confirm?

ETAS / (E * T * A * S) = SEAT - SATE

All 4 lettes are separate digits from 1 to 9.

ETAS, SEAT and SATE are 4 digit numbers

The r.h.s. can be written $$EAT-ATE = 100 \times E + AT - (10 \times AT + E) = 9 \times (EE - AT)$$ so $$9$$ must divide the digit sum $$A+E+S+T$$. Also the smallest product satisfying this constraint is $$1 \times 2 \times 6 \times 9 = 108$$ so $$ETAS \ge 972 \times (EE-AT)$$, therefore $$EE-AT$$ must be single-digit. Hence $$E=A+1$$ and $$EE-AT>E$$, in fact, it must be $$EE-AT=E+1$$, thus $$T=9$$, and $$9 \times A \times E \times S \times T < 1000$$. This leaves only $$1,2,6,9$$.
$$2916 / (2\times 9\times\ 1\times 6) = 6219 - 6192$$