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**