This is a lecture about Euler’s Proposition, which is the second step in a well-known proof of Gauss’s Theorem on Quadratic Reciprocity. We try to explain the proof of Euler’s Proposition using concrete numbers. As an example, if p=7 and a=4, then 4a=16. Consider that q=23 is equivalent to 7 mod 16, and 41 is equivalent to -7 mod 16. Then Euler’s Proposition ensures the Legendre symbols (4/7), (4/23), and (4/41) are all the same, in this case they’re all 1.