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silent hill 2 remake xbox - Tacotoon

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Thus, all 30 factor pairs produce valid lattice points.

📌 Hence, the total number of lattice points is $ \boxed{30} $.

📌 Question: What is the sum of all the positive divisors of $ 1440 $ that are congruent to $ 1 \pmod{4} $?

📌 Solution: First, factor $ 1440 $:

📌 = 144 \cdot 10 = (12^2) \cdot 2 \cdot 5 = (2^4 \cdot 3^2) \cdot 2 \cdot 5 = 2^5 \cdot 3^2 \cdot 5

📌 We seek the sum of all positive divisors $ d \mid 1440 $ such that $ d \equiv 1 \pmod{4} $.

📌 A divisor $ d $ satisfies $ d \equiv 1 \pmod{4} $ if it is odd (so $ d $ not divisible by 2) and $ d \equiv 1 \pmod{4} $. So restrict to odd divisors. Since $ d $ must be odd, we ignore the power of 2. So consider only divisors of $ 3^2 \cdot 5 = 45 $.

📌 The odd divisors of 1440 are exactly the divisors of 45. List them:

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