text: # Simple Proof
  
  bomba923
  Where $$\mathbb{Z}^{+}$$ represents the set of all positive integers,
  How do I prove that
  
  $$\begin{gathered} \forall \left\{ {a_0 ,a_1 ,a_2 , \ldots ,a_n } \right\} \subset \mathbb{Z}^ + \;{\text{where}}\;\max \left\{ {a_0 ,a_1 ,a_2 , \ldots ,a_n } \right\} \leqslant 9, \hfill \\ \left( {\sum\limits_{k = 0}^n {a_k 10^k } } \right)\;{\text{is divisible by }}3{\text{ iff }}\left( {\sum\limits_{k = 0}^n {a_k } } \right)\;{\text{is divisible by 3}} \; {?} \hfill \\ \end{gathered}$$
  
  Last edited:
  
  Staff Emeritus
  Gold Member
  Generally, you would want to look at problems like this modulo 3. Or, equivalently, to see if 3 divides their difference.
  
  bomba923
  Do you know any good books or sources on modulus?
  (I'm only a HS student, just started CalcIII)
  
  Homework Helper
  Gold Member
  I used induction and the fact that if an integer z is divisible by 3, then there exists (a unique) integer m such that z = 3m. Strangely though, I did not use the fact that the a_i's are smaller or equal to 9...
  
  Last edited:
  eok20
  Consider that a number is divisible by 3 iff the digits of the number add up to a multiple of three and that multiplying a number by a multiple of 10 just adds 0s to it and thus doesn't affect the sum of its digits.
  
  Muzza
  Consider that a number is divisible by 3 iff the digits of the number add up to a multiple of three
  
  This is equivalent to bomba923's question.