Suppose p is an odd prime and a 2 Z is not divisible by p. It turns out that 1 always has exactly 2 square roots in Z/(p) for odd prime p: 1 and −1. (a) Please prove that a has a square root in Z/(p) if and only if a(p−1)/2 =1 mod p. Hint: The "only if" direction (a has a square root in Z/(p) =) a(p−1)/2=1) is very easy to prove if you remember FLIT. To prove the "if" direction you may assume the basic fact that (Z/(p)) always has a generator. How then, would you express a square root of a if there is one? (b) Please prove that if p=3 mod 4 and a has a square root mod p, then the square roots of a are exactly a(p+1)/4 and −a(p+1)/4. Hint: To show that a as no more than 2 square roots you’ll need to use the fact that a quadratic polynomial has at most 2 roots in Z/(p). We’ll say more about this fact later in class.

Atrain traveled for 1.5 hours to the first station, stopped for 30 minutes, then traveled for 4 hours to the final station where it stopped for 1 hour. the total distance traveled is a function of time. which graph most accurately represents this scenario? a graph is shown with the x-axis labeled time (in hours) and the y-axis labeled total distance (in miles). the line begins at the origin and moves upward for 1.5 hours. the line then continues upward at a slow rate until 2 hours. from 2 to 6 hours, the line continues quickly upward. from 6 to 7 hours, it moves downward until it touches the x-axis a graph is shown with the axis labeled time (in hours) and the y axis labeled total distance (in miles). a line is shown beginning at the origin. the line moves upward until 1.5 hours, then is a horizontal line until 2 hours. the line moves quickly upward again until 6 hours, and then is horizontal until 7 hours a graph is shown with the axis labeled time (in hours) and the y-axis labeled total distance (in miles). the line begins at the y-axis where y equals 125. it is horizontal until 1.5 hours, then moves downward until 2 hours where it touches the x-axis. the line moves upward until 6 hours and then moves downward until 7 hours where it touches the x-axis a graph is shown with the axis labeled time (in hours) and the y-axis labeled total distance (in miles). the line begins at y equals 125 and is horizontal for 1.5 hours. the line moves downward until 2 hours, then back up until 5.5 hours. the line is horizontal from 5.5 to 7 hours