# Math Help - Prime Divisability

1. ## Prime Divisability

Let p be an odd prime number. Prove that the numerator of 1+1/2+1/3+...+1/(p-1) (when expressed as a single fraction with denominator (p-1)!) is divisible by p.

2. Originally Posted by chiph588@
Let p be an odd prime number. Prove that the numerator of 1+1/2+1/3+...+1/(p-1) (when expressed as a single fraction with denominator (p-1)!) is divisible by p.
Let $N$ be numerator.

Then, $N\equiv \sum_{k=1}^{p-1} p! (k)^{-1} (\bmod p)$. Where $(k)^{-1}$ is inverse mod $p$.

But $\sum_{k=1}^{p-1} p! (k)^{-1} = p! \sum_{k=1}^{p-1}k$.
Since $(k)^{-1}$ is a permuation of $\{1,2,...,p-1\}$.

Finally, $\sum_{k=1}^{p-1} k = \frac{p(p-1)}{2}$.
Therefore, it is divisible by $p$.