I thought I’d add one more thing:
You can construct your own numbers of this type by playing around with 'em. For example, Sublight was definitely on the right track with one of his remarks, about the number ending in either 17, 24, 55, 79 or 93.
Let’s assume the number ends in 3 (other choices may work as well, but we now know they won’t give you the smallest number). As Sublight mentioned, this forces the next digit to be 9 (3*3=9).
93*3 = 279, so the next digit must be 7.
793*3 = 2379, so the next digit is 3.
3793*3=11379, so the next digit is 1.
13793*3=41379, so the next digit is 4.
And so on…
Stop it when you get a number that works.
For another example, say, “What is the smallest number where if you move the last digit to the first (in other words 1234 becomes 4123) the new number is 2 times the size of the original number?”
I happen to know the smallest such number ends in 2 (but try another number if you like):
2*2=4, so the next digit is 4.
42*2=84, so the next digit is 8.
842*2=1684.
6842*2=13684
36842*2=73684
736842*2=1473684
4736842*2=9473684
94736842*2=189473684
894736842*2=1789473684
7894736842*2=15789473684
57894736842*2=115789473684
157894736842*2=315789473684
3157894736842*2=6315789473684
63157894736842*2=126315789473684
263157894736842*2=526315789473684
5263157894736842*2=10526315789473684
105263157894736842*2 = 210526315789473684