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Home » \( \textDistance = 60 \text miles/hour \times 2.5 \text hours = 150 \text miles \). - Dyverse

\( \textDistance = 60 \text miles/hour \times 2.5 \text hours = 150 \text miles \). - Dyverse

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 144y^2 + 3600 \equiv 0 \pmod{25} \Rightarrow 144y^2 \equiv -3600 \pmod{25}.

📌 Since $144 \equiv 19 \pmod{25}$ and $-3600 \equiv 0 \pmod{25}$, we get:

📌 19y^2 \equiv 0 \pmod{25} \Rightarrow y^2 \equiv 0 \pmod{25} \Rightarrow y \equiv 0 \pmod{5}.

📌 So let $y = 5k$. Substitute:

📌 25x^2 = 144(25k^2) + 3600 = 3600k^2 + 3600 = 3600(k^2 + 1).

📌 Divide both sides by 25:

📌 x^2 = 144(k^2 + 1).

📌 So $x^2 = 144(k^2 + 1) = (12)^2(k^2 + 1)$, hence $x = 12\sqrt{k^2 + 1}$. For $x$ to be integer, $k^2 + 1$ must be a perfect square. Let $k^2 + 1 = m^2$, so:

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