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This problem is taken from Niven's textbook, 3.6.16. Prove that if $(a,p)=1$ and $p$ is an odd prime, then $\sum_{n=1}^{p}\left(\frac{an+b}{p}\right)=0$, where $\left(\frac{x}{y}\right)$ is the Legendre/Jacobi symbol. I feel that somehow, I have to show that the number of $-1$ must be the same of the number of $1$, but other than that, I don't really know how to tackle the problem. I did try a few examples, but there is no pattern as $b$ changes.

Martin Sleziak
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Huy Truong
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    See also http://math.stackexchange.com/questions/26212/legendre-symbol-showing-that-sum-m-0p-1-left-fracambp-right-0 http://math.stackexchange.com/questions/1530467/sum-of-legendre-symbols-sum-n-1p-left-fracanbp-right-0 – Martin Sleziak Feb 22 '16 at 14:12

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Hint: As $n$ ranges from $1$ to $p$, $an+b$ ranges over all residue classes modulo $p$.

André Nicolas
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