Proving Divisibility by 3 for $\mathbb{Z}^{+}$ Sets

In summary, the conversation discusses how to prove that a sum of numbers in the set of positive integers, where the largest number is less than or equal to 9, is divisible by 3 if and only if the sum of the numbers is divisible by 3. The conversation also mentions the use of induction and the fact that multiplying a number by a multiple of 10 does not affect the sum of its digits. This is related to modulus and there are likely resources available for further understanding.
  • #1
bomba923
763
0
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]
 
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  • #2
Generally, you would want to look at problems like this modulo 3. Or, equivalently, to see if 3 divides their difference.
 
  • #3
:redface: Do you know any good books or sources on modulus?
(I'm only a HS student, just started CalcIII)
 
  • #4
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...
 
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  • #5
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.
 
  • #6
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.
 

1. How do you prove divisibility by 3 for positive integers?

To prove divisibility by 3 for positive integers, we can use the rule that a number is divisible by 3 if the sum of its digits is also divisible by 3. For example, the number 123 is divisible by 3 because 1+2+3=6, which is divisible by 3. This rule can be applied to any positive integer.

2. Why is it important to understand divisibility by 3?

Understanding divisibility by 3 is important because it is a fundamental concept in mathematics and can be applied to various mathematical problems. It also helps in simplifying calculations and identifying patterns in numbers.

3. Can you explain the concept of divisibility by 3 using a real-life example?

Yes, for example, if you have a jar of 18 cookies and want to divide them equally among 3 friends, you can use the concept of divisibility by 3 to determine that each friend will get 6 cookies. This is because 18 is divisible by 3.

4. Are there any other methods to prove divisibility by 3 besides the digit sum rule?

Yes, there are other methods such as using the divisibility rule for 3, which states that a number is divisible by 3 if the sum of its digits is divisible by 3. Another method is to use prime factorization, where if a number has a prime factor of 3, it is divisible by 3.

5. Can divisibility by 3 be extended to other sets of numbers besides positive integers?

Yes, the concept of divisibility can be extended to other sets of numbers such as negative integers, fractions, and decimals. However, the rules and methods may vary slightly for each set. For example, in the case of fractions, a fraction is divisible by 3 if both the numerator and denominator are divisible by 3.

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