Chris labels every lattice point in the coordinate plane with the square of the distance from the point to the origin (a lattice point is a point such that both of its coordinates are integers). How many times does he label a point with the number 25?
(it's not 4) please put an explanation so i can learn from it. Thanks!
+35 Here's my solution:
We know that the points that he labels with the number 25 has a distance of sqrt(25) = 5 from the origin. Thus, using the distance formula and calling the points that satisfy the condition (x, y), we have:
sqrt(x^2+y^2) = 5. Sqaring both sides gives us x^2+y^2 = 25. See if you can solve it from there! 
Hint: We have to find the number of points (x, y) that satisfy the equation x^2+y^2 = 25. We also have to remember that x and y can be negative numbers too, but they all have to be real integers. Listing the possibilities out, we get
(x, y) = (0, 5), (5, 0), (0, -5), (-5, 0), (3, 4), (-3, 4), (3, -4), (-3, -4), (4, 3), (-4, 3), (4, -3), and (-4, -3). How many pairs is that?
