A lattice point \((x, y)\) satisfies both \(x, y \in \mathbb{Z}\) and \(xy = 100\). We seek integer solutions to this equation.
["Finding All Integer Lattice Points ((x, y)) Such That (xy = 100)", "Understanding how to find integer solutions to mathematical equations is fundamental in number theory and algebra. One classic example involves lattice points in the coordinate plane—specifically, finding all pairs of integers ((x, y)) such that their product equals 100. This article explores how to systematically determine all such lattice points.", "---", "### What Are Lattice Points?", "A lattice point refers to a point in the coordinate plane where both coordinates are integers. In this context, we seek all integer coordinate pairs ((x, y)) satisfying:", "[\nxy = 100\n]", "with (x, y \in \mathbb{Z}).", "Since both (x) and (y) are integers, this equation defines a hyperbola. However, due to the integer constraint, solutions are discrete and finite.", "---", "### Step 1: Analyze the Equation", "We are solving:", "[\nxy = 100\n]", "Since (x) and (y) are integers, (x) must be a divisor of 100. Thus, every integer divisor (d) of 100 gives rise to a solution:", "[\nx = d, \quad y = \frac{100}{d}\n]", "Therefore, each integer divisor of 100 corresponds to a lattice point ((d, 100/d)).", "---", "### Step 2: Find All Integer Divisors of 100", "First, factorize 100:", "[\n100 = 2^2 \ imes 5^2\n]", "The number of positive divisors is:", "[\n(2+1)(2+1) = 9\n]", "So, 100 has 9 positive divisors and 9 negative divisors.", "Positive divisors of 100:\n1, 2, 4, 5, 10, 20, 25, 50, 100", "Negative divisors of 100:\n-1, -2, -4, -5, -10, -20, -25, -50, -100", "---", "### Step 3: List All Integer Pairs ((x, y))", "For each divisor (x = d), compute (y = 100/d). Since (d) can be positive or negative, all 18 divisors yield distinct lattice points.", "List of solutions:", "[\n(1, 100),\ (2, 50),\ (4, 25),\ (5, 20),\ (10, 10),\ (20, 5),\ (25, 4),\ (50, 2),\ (100, 1)\n]\n[\n(-1, -100),\ (-2, -50),\ (-4, -25),\ (-5, -20),\ (-10, -10),\ (-20, -5),\ (-25, -4),\ (-50, -2),\ (-100, -1)\n]", "These are all 18 integer lattice points on the hyperbola (xy = 100), both in the first and third quadrants (where both coordinates have the same sign).", "Note: Since (xy = 100 > 0), both (x) and (y) have the same sign—so solutions occur only in quadrants I and III.", "---", "### Step 4: Avoid Duplicate Solutions", "Each pair combines a unique divisor with its quotient. No duplicates arise because each divisor maps uniquely to a (y) value. Thus, we have exactly 18 distinct lattice points.", "---", "### Step 5: Mathematical Insight – Symmetry and Factorization", "This problem illustrates a key principle: solutions to (xy = n) in integers correspond to divisors of (n). Each divisor pair ((d, n/d)) generates a lattice point. The totality of solutions is symmetric with respect to the origin but confined to quadrants where sign consistency holds.", "For (xy = 100), the structure of the divisor set ensures completeness and finiteness—only 18 valid integer pairs exist.", "---", "### Conclusion", "Identifying lattice points satisfying (xy = 100) relies entirely on integer divisor pairs. By enumerating all integer divisors of 100—and assigning each to a corresponding quotient—we systematically uncover all 18 integer lattice points on this hyperbolic curve. This method exemplifies how number theory and algebra intersect to solve elegant geometric problems.", "Whether used in cryptography, physics, or pure mathematics, understanding how to generate lattice points from multiplicative equations remains a foundational skill.", "---", "Keywords: lattice point, integer solutions, (xy = 100), divisor pairs, integer coordinates, number theory, factorization, hyperbola lattice points", "Meta description: Discover all integer lattice points ((x, y)) such that (xy = 100). Learn how to find them using divisor pairs and number theory. Full guide on lattice solutions with examples and explanation."]
