How many lattice points lie on the line segment joining (1, 2) and (7, 8)?
["# How Many Lattice Points Lie on the Line Segment Joining (1, 2) and (7, 8)?", "When studying geometry and number theory together, one fascinating question is: How many lattice points—points with integer coordinates—lie on a straight line segment connecting two points in the coordinate plane? A classic and insightful example involves finding the number of lattice points on the line segment joining (1, 2) and (7, 8).", "This article explores this geometric problem using number theory, particularly the properties of the greatest common divisor (GCD), offering both clarity and practical computation.", "---", "## What Is a Lattice Point?", "A lattice point is any point ((x, y)) in the plane where both (x) and (y) are integers. For example, (0, 0), (3, 5), and (−1, 4) are lattice points, but (2.5, 3) is not.", "---", "## The Line Segment from (1, 2) to (7, 8)", "We are examining the straight line segment connecting the points (A = (1, 2)) and (B = (7, 8)). Since both endpoints are lattice points, many lines connecting them contain lattice points in between — but how many exactly?", "This problem is fundamentally related to the greatest common divisor (GCD) of the differences in coordinates. Specifically, if we consider the vector from (A) to (B), the number of lattice points inclusive on the segment depends exactly on (\gcd(\Delta x, \Delta y)).", "---", "## Step 1: Compute the Differences", "First, compute the differences in the x- and y-coordinates:", "[\n\Delta x = 7 - 1 = 6,\quad \Delta y = 8 - 2 = 6\n]", "So the displacement vector is (6, 6), but the key mathematical object is (\Delta x) and (\Delta y) relative to the segment.", "---", "## Step 2: Apply the Lattice Points Formula", "The number of lattice points (including endpoints) on the line segment joining two lattice points ((x_1, y_1)) and ((x_2, y_2)) is given by:", "[\n\gcd(|x_2 - x_1|, |y_2 - y_1|) + 1\n]", "This formula works because the line inherently divides the “grid” evenly at intervals determined by the GCD.", "For our points:", "[\n\gcd(|7 - 1|, |8 - 2|) = \gcd(6, 6) = 6\n]", "Therefore, the total number of lattice points on the segment is:", "[\n6 + 1 = 7\n]", "---", "## Step 3: Verify by Finding All Points", "Let’s confirm this by identifying the lattice points explicitly.", "The parametric equations of the line segment from (t = 0) to (t = 1) are:", "[\nx = 1 + 6t,\quad y = 2 + 6t,\quad t \in [0,1]\n]", "We seek values of (t) such that both (x) and (y) are integers.", "Let (t = \frac{k}{d}), where (d = \gcd(6, 6) = 6), and (k = 0, 1, 2, \dots, 6):", "[\nx = 1 + 6\cdot\frac{k}{6} = 1 + k,\quad y = 2 + 6\cdot\frac{k}{6} = 2 + k\n]", "So the lattice points are:", "[\n(1 + k, 2 + k),\quad k = 0, 1, 2, 3, 4, 5, 6\n]", "That gives:\n(1,2), (2,3), (3,4), (4,5), (5,6), (6,7), (7,8) — exactly 7 points.", "---", "## Why Does the GCD Determine the Count?", "The line from (1,2) to (7,8) has slope 1, and increases equally in both x and y. Every time (t) increases by a fraction where both numerator and denominator divide evenly into the step size (6), the coordinates remain integers.", "The number of such valid intermediate steps corresponds precisely to the number of common divisors of the run and rise—here, 6 and 6, giving 6 one-unit steps—and the +1 accounts for the starting point.", "---", "## Conclusion", "The line segment from (1, 2) to (7, 8) contains exactly 7 lattice points, including the endpoints. This result exemplifies a powerful principle in discrete geometry: the number of lattice points on a line segment depends elegantly on the GCD of the coordinate differences.", "Understanding this connection helps in solving various problems in computational geometry, crystal lattice modeling, and digital mapping.", "---", "## SEO Optimization Notes", "- Key keywords: lattice points, line segment geometry, (\gcd) formula, count lattice points, integer coordinates, discrete geometry\n- Long-tail keywords: how many lattice points on line segment from (1,2) to (7,8), number of lattice points on straight line in grid\n- Structure: Clear headline, step-by-step explanation, verification with examples, a logical flow from definition to proof\n- User intent: Answer a precise geometrical question with both conceptual insight and exact calculation, ideal for learners, students, and educators.", "---", "Understanding how many lattice points lie on a line segment enhances comprehension of number theory and coordinate geometry. Mastery of such concepts supports deeper learning in mathematical modeling and algorithmic geometry."]
