Problem 11 · AMC 8 Stretch
Stretch
Number Theory
Geometry & Measurement
logical-reasoningpattern-recognition
A lattice point has both coordinates whole numbers, like \((4, 3)\). A straight line through the origin passes through the lattice point \((6, 4)\). What is the lattice point on this line that is CLOSEST to the origin (other than the origin itself)?
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Answer: (3, 2)
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Hint 1 of 4
Find the slope of the line through \((0,0)\) and \((6, 4)\): rise over run is \(\tfrac{4}{6}\). Reduce it.
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Hint 2 of 4
\(\tfrac{4}{6} = \tfrac{2}{3}\) (go right 3, up 2). Once a line through the origin hits one lattice point, it hits all whole-number multiples of it.
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Hint 3 of 4
From \((0,0)\) step right 3, up 2 to land on a lattice point. What is the first one you reach?
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Approach: Reduce the slope to lowest terms
- The line through \((0,0)\) and \((6,4)\) has slope \(\tfrac{4}{6} = \tfrac{2}{3}\): go right 3, up 2.
- A line through the origin passes through every whole-number multiple of a lattice point on it: \((3,2), (6,4), (9,6), (12,8), \dots\)
- The closest one to the origin is the smallest step, \((3, 2)\), where the slope \(\tfrac{2}{3}\) is already in lowest terms.
- So the closest lattice point is \((3, 2)\).
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