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

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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

$\gcd \set {a_0, a_1, \ldots, a_n}$ is divisible by $d$.

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

Proof

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