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Home » \( T = 3 \, \textسنوات \) (عمر النصف). - Coaching Toolbox

\( T = 3 \, \textسنوات \) (عمر النصف). - Coaching Toolbox

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 Thus, the largest possible value of \(\gcd(a, b)\) is \(\boxed{289}\).

📌 A game-based learning researcher, exploring geometric puzzles, asks: How many lattice points lie on the hyperbola \(xy = 100\)?

📌 A lattice point \((x, y)\) satisfies both \(x, y \in \mathbb{Z}\) and \(xy = 100\). We seek integer solutions to this equation.

📌 Let \(d\) be a positive divisor of 100. Then \(x = d\) and \(y = \frac{100}{d}\) is an integer. Since \(xy = 100\), both \((d, 100/d)\) and \((-d, -100/d)\) are solutions.

📌 First, factor 100:

📌 \(100 = 2^2 \times 5^2\), so the number of positive divisors is \((2+1)(2+1) = 9\).

📌 The positive divisors are: \(1, 2, 4, 5, 10, 20, 25, 50, 100\).

📌 Each positive divisor \(d\) gives a solution \((d, 100/d)\), and each negative divisor \(-d\) gives \((-d, -100/d)\).

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