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Math Help - Simple Proof

  1. #1
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    Simple Proof

    I feel like this proof should be easy.

    Prove that for 3 consecutive natural numbers, 1 of the numbers will be divisible by 3.

    Intuitively this makes sense (add 1, 2 or 3 and then it's divisible by 3, obviously).
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    let  n \in \mathbb{N}, A = \{n, n+1, n+2\}

    if  n \equiv 0 \mod{3} , then  A \equiv \{0,1,2\} \mod{3}

    if  n \equiv 1 \mod{3} , then  A \equiv \{1,2,0\} \mod{3}

    if  n \equiv 2 \mod{3} , then  A \equiv \{2,0,1\} \mod{3}

    So in all three cases,  \exists \; a \in A s.t.  a \equiv 0 \mod{3} \Rightarrow a is divisible by three.
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