Question 14

The graph of $$x^2 + y^2 = 25$$ is plotted in a real $$xy$$-plane. The number of points $$(x, y)$$ on this graph such that both $$x$$ and $$y$$ are integers, is

Solution

The only ways to write 25 as a sum of two squares are $$25 = 0 + 25 = 9 + 16$$. From $$0 + 25$$ we get the four points $$(\pm 5, 0)$$ and $$(0, \pm 5)$$, and from $$9 + 16$$ we get the eight points $$(\pm 3, \pm 4)$$ and $$(\pm 4, \pm 3)$$. Altogether there are 12 lattice points.

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