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assassin's creed hexe - Coaching Toolbox

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 Question: A green infrastructure project involves planting trees in a grid pattern across $ 1001 $ city blocks. If each block receives a number of trees equal to the number of positive divisors of its index (from $ 1 $ to $ 1001 $), what is the total number of trees planted modulo $ 10 $?

📌 Where $ d(n) $ is the number of positive divisors of $ n $. While computing the full sum is tedious, we can observe a known result:

📌 Solution: Let the four consecutive odd integers be $ n, n+2, n+4, n+6 $, where $ n $ is odd. Since they are consecutive odd numbers, they are all relatively prime in pairs. We analyze the divisibility properties:

📌 Among any four consecutive odd integers, at least one is divisible by $ 3 $, and at least one is divisible by $ 5 $.

📌 Since they are odd, none is divisible by $ 2 $, so we only consider odd prime factors.

📌 At least one of the four is divisible by $ 3 $

📌 Now consider $ 15 $. Since $ 3 $ and $ 5 $ always divide the product, the product is divisible by $ 15 $.

📌 But we can go further: among four consecutive odd numbers, at least one is divisible by $ 3 $, and at least one divisible by $ 5 $. Also, at least two of them are congruent modulo $ 4 $, so their product includes multiple factors of $ 2 $, but since all are odd, no power of $ 2 $ is introduced.

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