Solution: To find lattice points on the hyperbola $\frac{x^2}{144} - \frac{y^2}{25} = 1$, multiply both sides by $144 \cdot 25 = 3600$ to eliminate denominators:

["Finding Lattice Points on the Hyperbola: A Step-by-Step Solution Using Elimination of Denominators", "When analyzing conic sections like hyperbolas, identifying lattice points—points with integer coordinates—on their graph can be a challenging yet fascinating task. The hyperbola given by the equation\n$$\n\frac{x^2}{144} - \frac{y^2}{25} = 1\n$$\npresents a classic problem in number theory and geometry, especially when searching for integer solutions.", "### Why Eliminate Denominators?", "The equation of the hyperbola involves fractions, which can hinder direct searching for integer (lattice) points. Multiplying both sides of the equation by $144 \cdot 25 = 3600$ clears the denominators, simplifying the expression and transforming it into a more manageable form:", "$$\n3600 \left( \frac{x^2}{144} - \frac{y^2}{25} \right) = 3600 \cdot 1\n$$", "This simplifies neatly to:\n$$\n25x^2 - 144y^2 = 3600\n$$", "This integer equation—$25x^2 - 144y^2 = 3600$—now defines the same hyperbola in integer coefficients, making it easier to search systematically for all lattice points $(x, y)$ satisfying the original hyperbola.", "### Rewriting the Equation for Efficient Search", "Now, we seek all integer solutions $(x, y)$ such that:\n$$\n25x^2 - 144y^2 = 3600\n$$", "We can rearrange the equation:\n$$\n25x^2 = 144y^2 + 3600\n$$", "Or:\n$$\nx^2 = \frac{144y^2 + 3600}{25}\n$$", "For $x^2$ to be an integer, the numerator $144y^2 + 3600$ must be divisible by 25. We analyze this divisibility condition.", "### Divisibility Condition", "We require:\n$$\n144y^2 + 3600 \equiv 0 \pmod{25}\n$$", "Note that $144 \mod 25 = 19$, and $3600 \mod 25 = 0$, since $3600 = 25 \ imes 144$. So:\n$$\n19y^2 \equiv 0 \pmod{25}\n$$", "Since 19 and 25 are coprime, we can multiply both sides by the modular inverse of 19 modulo 25. The inverse of 19 modulo 25 is 16, because $19 \cdot 16 = 304 \equiv 1 \pmod{25}$. Thus:\n$$\ny^2 \equiv 0 \pmod{25}\n\Rightarrow 25 \mid y^2 \Rightarrow 5 \mid y\n$$", "Therefore, $y = 5k$ for some integer $k$. Substitute $y = 5k$ into the equation:", "$$\n25x^2 - 144(25k^2) = 3600\n\Rightarrow 25x^2 - 3600k^2 = 3600\n\Rightarrow 25x^2 = 3600(k^2 + 1)\n\Rightarrow x^2 = 144(k^2 + 1)\n$$", "For $x^2$ to be a perfect square, $144(k^2 + 1)$ must be a perfect square. Since 144 is already a perfect square ($12^2$), $k^2 + 1$ must also be a perfect square.", "Let:\n$$\nk^2 + 1 = m^2 \quad \ ext{for some integer } m\n$$", "Then:\n$$\nm^2 - k^2 = 1 \Rightarrow (m - k)(m + k) = 1\n$$", "The only integer solutions to this are $m = 1, k = 0$ or $m = -1, k = 0$. So the only solution is $k = 0$.", "### Determining the Lattice Points", "With $k = 0$, then $y = 5k = 0$. Substitute into the simplified equation:\n$$\n25x^2 = 3600 \Rightarrow x^2 = 144 \Rightarrow x = \pm 12\n$$", "Thus, the only lattice points on the hyperbola are:\n$$\n(12, 0) \quad \ ext{and} \quad (-12, 0)\n$$", "### Verification", "Check $(12, 0)$ in the original equation:\n$$\n\frac{12^2}{144} - \frac{0^2}{25} = \frac{144}{144} - 0 = 1 \quad \ ext{✓}\n$$", "Check $(-12, 0)$:\n$$\n\frac{(-12)^2}{144} - \frac{0^2}{25} = 1 \quad \ ext{✓}\n$$", "No other integer values of $k$ produce perfect squares for $x^2$, and hence no other lattice points exist.", "### Conclusion", "By multiplying through by $3600 = 144 \cdot 25$, we transformed a diophantine equation into a solvable form, leveraging divisibility and number theory to restrict possible solutions. This method illustrates a powerful strategy for finding lattice points on hyperbolas: eliminate denominators, analyze modular constraints, and reduce the problem to primitive divisibility conditions.", "For last-minute exams, olympiad problems, or learning purposes, mastering such techniques empowers deeper insight into geometric equations and enhances problem-solving flexibility.", "---", "Key Takeaways:\n- Clearing denominators simplifies conic section equations for integer solving.\n- Modular arithmetic helps drastically reduce search space.\n- Constraints like $y = 5k$ arise naturally from divisibility.\n- Linear Diophantine forms often hide deeper number-theoretic structure.", "ECSE: Apply algebraic transformation and number theory to solve for integer solutions on conic sections."]