divisibility - When an integer is disible by 17,19,23, or 41?

Let $n=a_m10^m++a_{m-1}10^{m-1}+\dots + a_{2}10^2+a_{1}10+a_0$ where $a_k$ are integers and $0\leq a_k \leq 9,k=0,1,\dots,m$ be the decimal representation of a positive integer $n$.
Let $S=a_0+a_1+\dots + a_m, T=a_0-a_1+ \dots + (-1)^m a_m$. Then
1.$n$ is divisible by $2$ iff a_0 is divisible by $2$.
2. $n$ is divisible by 11 iff $T$ is divisible by 11.
3. $n$ is divisible by 3 iff $S$ is divisible by 3.
I know the proof of these results. I shall prove first one and the proof of rest two are similar.
Now my question : Is there any similar rule which will tell us whether an integer
$n$ is divisible by the primes 17,19,23,37,41 etc or not?