Question

Determine the smallest prime that does not divide any five-digit number whose digits are in a strictly increasing order.

Answer 1

Suppose $12346$ is even, 3 and 5 divide $12345$, and 7 divides $12348$
Consider a 5 digit number $n=abcde$ with $0<a<b<c<d<e<10$
Let $S=(a+c+e)-(b+d)$
Then $S=a+\left(cb\right)+\left(ed\right)>a>0$ and $S=e-(d-c)-(b-a)<e\le 10$
So S is not divisible by 11 and hence n is not divisible by 11.
$\therefore$ 11 is the smallest prime that does not divide any five-digit number whose digits are in a strictly increasing order.