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Where [tex] \mathbb{Z}^{+}[/tex] represents the set of all positive integers,

How do I prove that

[tex] \begin{gathered} \forall \left\{ {a_0 ,a_1 ,a_2 , \ldots ,a_n } \right\} \subset \mathbb{Z}^ + \;{\text{where}}\;\max \left\{ {a_0 ,a_1 ,a_2 , \ldots ,a_n } \right\} \leqslant 9, \hfill \\

\left( {\sum\limits_{k = 0}^n {a_k 10^k } } \right)\;{\text{is divisible by }}3{\text{ iff }}\left( {\sum\limits_{k = 0}^n {a_k } } \right)\;{\text{is divisible by 3}} \; {?} \hfill \\ \end{gathered} [/tex]

How do I prove that

[tex] \begin{gathered} \forall \left\{ {a_0 ,a_1 ,a_2 , \ldots ,a_n } \right\} \subset \mathbb{Z}^ + \;{\text{where}}\;\max \left\{ {a_0 ,a_1 ,a_2 , \ldots ,a_n } \right\} \leqslant 9, \hfill \\

\left( {\sum\limits_{k = 0}^n {a_k 10^k } } \right)\;{\text{is divisible by }}3{\text{ iff }}\left( {\sum\limits_{k = 0}^n {a_k } } \right)\;{\text{is divisible by 3}} \; {?} \hfill \\ \end{gathered} [/tex]

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