Dividing by Dropping Digits

An integer decreases an integral number of times when its last D digits are dropped.  Obviously, there are infinite such integers, as any integer ending in D 0's decreases by a factor of 10^D when the training 0's are dropped.  Is the number of such integers finite, if the last D digits are no all 0's?  If it is finite, how many such integers exist for D=1,2,3,4?

Source: Sudipta Das.

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