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Find the number of lattice points in the region defined by |x| + |y| < 5.

 Dec 17, 2019
 #1
avatar+109752 
+1

See the graph here :  https://www.desmos.com/calculator/9riuxf41mb

 

We have 16 lattice points above and below the x axis and 9 points on the x axis....so....the total number of lattice points  =

 

16 + 16  + 9   =

 

41

 

 

cool cool cool

 Dec 17, 2019
 #2
avatar+41 
+3

The strategy here is to find the boundary of the graph and then figure out what the actual graph is. First, plot some points for |x| + |y| = 5 on the Cartersian plane; (5,0), (4,1), (3,2), and combinations of their opposites work as well (ex. -4,1).

 

The graph of |x| + |y| = 5 becomes a diamond centered at the origin with perpendicular diagonals of length 10 on the x and y-axis. 

 

If you think about it, |x| + |y| < 5 must be all of the area within the diamond.

 

At this point, you can count the number of lattice points by drawing a diagram. Be careful, as the inequality sign is less than, not less than or equal to, so do not count the lattice points on the diamond itself.

 

By counting with symmetry, you get

 

6*4 + 4*4 + 1 = 41. 

 

The 6*4 is for the points between the axes, the 4*4 is the points on the axes (excluding the origin), and the 1 is for the origin.

 

laugh good luck

 Dec 17, 2019

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