sum of the numbers less than p divisible by p?

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March 9th 2011, 09:42 AM
hmmmm

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
March 9th 2011, 10:26 AM
topspin1617

Quote:

Originally Posted by hmmmm

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):

$[1+(p-1)]+[2+(p-2)]+\cdots +[\frac{p-1}{2}+\frac{p+1}{2}]$
March 9th 2011, 10:40 AM
hmmmm

$\sum_{n=1}^{p-1}{n}=\frac{p-1}{2}*p$ as p-1 is even (so divisible by 2) then it is divisible by p?
March 9th 2011, 06:19 PM
topspin1617

Exactly. Writing the sum as I did above, you immediately get what you have just written.

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