# Question involving Legendre symbols

Let r,p,q be distinct odd primes. Let 4r divide p-q. Show that

(r/p) = (r/q)

Where (a/b) is the Legendre symbol.

I'm sure we are suppose to use the law of quadratic reciprocity. I don't think this question is suppose to be difficult, but I cannot figure it out!

• So what do you get when you apply QR to (r/p) for instance? You should at least play around some. Companion to Zev's hint: $$\begin{cases}p\equiv q\bmod r\implies \left(\frac{p}{r}\right)=\left(\frac{q}{r}\right) \\ p\equiv q\bmod 4\implies \frac{p-1}{2}\equiv\frac{q-1}{2}\bmod 2 \end{cases}$$
– anon
Commented Jun 2, 2013 at 7:35

Hint: $$4r\mid (p-q)\iff p\equiv q\bmod r\quad\text{and}\quad p\equiv q\bmod 4$$

• This means (p/r)=(q/r), right? Commented Jun 2, 2013 at 7:39
• @John: Just $p\equiv q\bmod r$ alone is enough to know that $(\frac{p}{r})=(\frac{q}{r})$. Do you see how to use the other piece of information? Commented Jun 2, 2013 at 7:41
• Yes! Thank you so much. Commented Jun 2, 2013 at 7:45
• Glad I could help! Commented Jun 2, 2013 at 7:51