This is obviously an "alphametic",
. . where and represent 3-digit numbers.
I began an orderly (but lengthy) search
. . and virtually tripped over the solution.
The problem has this form:
Suppose , then
Then we have:
The suspicious "shape" of the two factors: .
. . led me to try first . . . and it worked!
. . . . .
Multiplication is commutative, which means that the order you multiply is unimportant - you'll get the same result regardless.
Therefore, ABC = BCA = CAB = CBA
If you divide any number by itself, you get 1:
ABC and CBA cancel out, to give:
Which, of course, makes no sense, so ABC must equal 0. They must all be zero.
EDIT: Sorry, I missed your post and assumed they were variables... my mistake! I'll leave this here anyway.