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Math Help - sum of the numbers less than p divisible by p?

  1. #1
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    sum of the numbers less than p divisible by p?

    is  \sum_{n=1}^{p-1}{n} always divisible by p if p is prime?

    Thanks for any help
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  2. #2
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    Quote Originally Posted by hmmmm View Post
    is  \sum_{n=1}^{p-1}{n} always divisible by p if p is prime?

    Thanks for any help
    Yes, unless p=2. In fact, it is true as long as p is ANY odd number.

    Try rewriting the sum (for p>2, in which case p is odd):

    [1+(p-1)]+[2+(p-2)]+\cdots +[\frac{p-1}{2}+\frac{p+1}{2}]
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  3. #3
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     \sum_{n=1}^{p-1}{n}=\frac{p-1}{2}*p as p-1 is even (so divisible by 2) then it is divisible by p?
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  4. #4
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    Exactly. Writing the sum as I did above, you immediately get what you have just written.
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