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Math Help - Prove n is divisible by 3

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    Prove n is divisible by 3

    Prove that if n is a positive integer such that the sum of the digits of n in decimal representation is divisible by 3 then n is divisible by 3
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    Quote Originally Posted by mandy123 View Post
    Prove that if n is a positive integer such that the sum of the digits of n in decimal representation is divisible by 3 then n is divisible by 3
    This one is sort of the archetype for the rest of them.

    Note that
    10^n \equiv 1 \text{ mod 3}
    for all positive n.

    So putting a number x into its decimal representation in base 10:
    x = \sum_n x_n \cdot 10^n

    We see that
    x \equiv \sum_n x_n \cdot 1 \text{ mod 3}

    So if \sum_n x_n = 0 \text{ mod 3} (ie. the sum of the digits of x is divisible by 3), then the number x is divisible by 3.

    -Dan
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