Originally Posted by

**ilikepizza** Also I think the 3rd number is 4, just not sure how to explain it.

Any ideas for what the smallest number could be?

The third number is $\displaystyle 4$ but only if you want to use the digit $\displaystyle 9.$ We already know that $\displaystyle 0$ and $\displaystyle 5$ cannot occur in our number; let's show that $\displaystyle 4$ and $\displaystyle 9$ cannot both occur in our number. Suppose they do. A number is divisible by $\displaystyle 9$ if and only if the sum of its digits is divisible by $\displaystyle 9$ so let's check for divisibility by $\displaystyle 9$ for the candidates for our number:

$\displaystyle 9 + 8 + 7 + 6 + 4 + 3 + 2 = 39$

$\displaystyle 9 + 8 + 7 + 6 + 4 + 3 + 1 = 38$

$\displaystyle 9 + 8 + 7 + 6 + 4 + 2 + 1 = 37$

$\displaystyle 9 + 8 + 7 + 4 + 3 + 2 + 1 = 34$

$\displaystyle 9 + 8 + 6 + 4 + 3 + 2 + 1 = 33$

$\displaystyle 9 + 7 + 6 + 4 + 3 + 2 + 1 = 32$

None of these is divisible by $\displaystyle 9.$ Hence if we use the $\displaystyle 4$ we can't use the $\displaystyle 9$ – but we do want to use the $\displaystyle 9$ so as to make our number as large as possible.