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august 2025 calendar printable - Deep Underground Poetry

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 We analyze divisibility by small primes:

📌 Divisibility by 3**: Among any four consecutive odd integers, at least one is divisible by 3. This can be shown by checking modulo 3: the residues of odd integers mod 3 are 0, 1, or 2. In any block of 6 consecutive integers, four odd ones must include one divisible by 3. So $3 \mid P$.

📌 Divisibility by 5**: Not guaranteed — e.g., $n = 1$: $1 \cdot 3 \cdot 5 \cdot 7 = 105$, divisible by 5, but $n = 7$: $7 \cdot 9 \cdot 11 \cdot 13 = 9009$, not divisible by 5. So 5 does not always divide $P$.

📌 Divisibility by higher powers or other primes?

📌 Check powers of 3: Consider $n = 3$: $3 \cdot 5 \cdot 7 \cdot 9 = 945 = 3^3 \cdot 5 \cdot 7$ — divisible by $27$.

📌 But $n = 9$: $9 \cdot 11 \cdot 13 \cdot 15 = 19305 = 3^3 \cdot 5 \cdot 11 \cdot 13$ — divisible by $27$.

📌 But $n = 15$: $15 \cdot 17 \cdot 19 \cdot 21 = 3 \cdot 5 \cdot 17 \cdot 3 \cdot 7 \cdot 3 \cdot 7 = 3^3 \cdot 5 \cdot 7^2 \cdot 17$ — divisible by $27$.

📌 But not always divisible by $81$ — try $n = 1$: only $3^1$. So maximum power of 3 is $3^3$.

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