2000 AIME II Problems/Problem 2

Problem

A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2 - y^2 = 2000^2$ ?

Solution

$(x-y)(x+y)=2000^2=2^8*5^6$

We must have $(x-y)$ and $(x+y)$ as both even or else x,y would not be an integer. We first give a factor of two to both $(x-y)$ and $(x+y)$ . We have $2^6*5^6$ left. Since there are $7*7=49$ factors of $2^6*5^6$ , and since both x and y can be negative, this gives us $49\cdot2=98$ lattice points.

2000 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
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2000 AIME II Problems/Problem 2

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