Interesting problem

145 does the trick.

if you just look 7! is more that 3 digits, so none of the numbers can be 7.

which means the number would have to have at least a 7 in the first place, which means it is too big by above.

This narrows your choices to 1,2,3,4,5.

If you use all 4 or lower its not enough to get up into the 3 digits.

So you know there has to be a 5 in there somewhere. From there I just played around with it.