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
This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |