Numbers with All Digits Have a Common Factor are Divisible by This Factor

Theorem
A number expressed in decimal notation is divisible by $d$ if all its digits are divisible by $d$.

That is:
 * $N = \sqbrk {a_0 a_1 a_2 \ldots a_n}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $d$

if
 * $gcd(a_0,a_1,\ldots,a_n)$ is divisible by $d$.

This theorem is in fact true in all base $b$.