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Home » So $d = 1012$ is achievable. - AMAZONAWS

So $d = 1012$ is achievable. - AMAZONAWS

📅 March 5, 2026 👤 scraface

Mar 05, 2026

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📌 For \(x\) and \(y\) to be integers, \(a + b\) and \(b - a\) must both be even, so \(a\) and \(b\) must have the same parity.

📌 Now factor 2024:

📌 = 2^3 \cdot 11 \cdot 23

📌 The total number of positive divisors is \((3+1)(1+1)(1+1) = 16\). So there are 16 positive factor pairs \((a, b)\) with \(ab = 2024\), and 16 negative ones (since \(a, b\) both negative gives the same product), for a total of 32 factor pairs.

📌 But only pairs where \(a\) and \(b\) have the same sign and same parity contribute valid solutions.

📌 For \(x\) and \(y\) to be integers, \(a\) and \(b\) must both be even (since if both were odd, \(ab\) would be odd; if one odd and one even, \(a + b\) is odd). So both \(a\) and \(b\) must be even.

📌 Let \(a = 2m\), \(b = 2n\), then \(4mn = 2024 \Rightarrow mn = 506\).

📌 Now \(506 = 2 \cdot 11 \cdot 23\), so it has \((1+1)(1+1)(1+1) = 8\) positive divisors, giving 8 positive and 8 negative pairs \((m, n)\).

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