Therefore, the total number of lattice points on \(xy = 100\) is \(\boxed{18}\).
["Therefore, the Total Number of Lattice Points on (xy = 100) is (\boxed{18})", "Lattice points—points with integer coordinates—are fundamental in number theory, geometry, and discrete mathematics. One classic problem involves determining how many lattice points lie on the hyperbola defined by the equation (xy = 100). This article explains why the total number of such points is exactly 18, combining insights from number theory and coordinate geometry.", "### What Are Lattice Points on (xy = 100)?", "A lattice point ((x, y)) lies on the curve (xy = 100) if both (x) and (y) are integers satisfying the equation. Since (xy = 100), every positive integer divisor (d) of 100 gives a solution ((d, 100/d)), and each such pair and its reflection ((-d, -100/d)) also satisfies the equation.", "### Step 1: Count Positive Integer Solutions", "First, find all positive integer pairs ((x, y)) such that (xy = 100). This requires identifying all positive divisors of 100.", "Factor 100 into primes:\n[\n100 = 2^2 \ imes 5^2\n]", "The number of positive divisors of 100 is:\n[\n(2+1)(2+1) = 9\n]", "The positive divisors are:\n[\n1, 2, 4, 5, 10, 20, 25, 50, 100\n]", "Each divisor (d) gives a solution ((d, 100/d)). So there are 9 positive lattice points:\n[\n(1,100), (2,50), (4,25), (5,20), (10,10), (20,5), (25,4), (50,2), (100,1)\n]", "### Step 2: Count Negative Integer Solutions", "Because the equation (xy = 100) is symmetric under sign changes (i.e., ((x, y)) and ((-x, -y)) both satisfy it), for every positive solution ((d, 100/d)), the pair ((-d, -100/d)) is also a valid lattice point.", "Thus, there are 9 corresponding negative solutions:\n[\n(-1,-100), (-2,-50), (-4,-25), (-5,-20), (-10,-10), (-20,-5), (-25,-4), (-50,-2), (-100,-1)\n]", "### Step 3: Total Number of Lattice Points", "Adding both sets:\n- 9 positive lattice points\n- 9 negative lattice points", "Total:\n[\n9 + 9 = \boxed{18}\n]", "### Why Exclude Mixed Sign Points?", "One might wonder if mixed sign pairs, such as ((d, -100/d)), lie on the curve. However, substituting into (xy = 100) gives (d \cdot (-100/d) = -100 <br/>\ne 100), so these do not satisfy the equation. Only points where both coordinates are positive or both are negative work.", "### Conclusion", "The total number of lattice points on the hyperbola (xy = 100) is precisely 18, consisting of 9 points above the origin and 9 below. This elegant result arises directly from the factorization of 100 and the symmetry of the equation. Understanding such problems strengthens skills in number theory and discrete geometry.", "Therefore, the total number of lattice points on (xy = 100) is (\boxed{18})."]
