In how many positive four-digit integers that are not multiples of 1111 do the digits form a geometric sequence? (The digits must form a geometric sequence, in order. For example, the number 8412 does not count.)

There are 8 numbers that work.

The sequence 1248 is geometric with a common ratio of 2.

The sequence 8421 is geometric with a common ratio of 1/2.

Each of the following are geometric, with a common ratio of 0:

1000 2000 3000 4000 5000 6000 7000 8000 9000.