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Home » Average = $rac14 \cdot rac9283 = rac92812 = 77.\overline3$. - Crosslake

Average = $rac14 \cdot rac9283 = rac92812 = 77.\overline3$. - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 \div 7 = 17 \text{ remainder } 1 \Rightarrow \text{Yes}

📌 Thus, the smallest such number is $\boxed{120}$.

📌 Question: What is the largest integer that must divide the product of any four consecutive odd integers?

📌 Solution: Let the four consecutive odd integers be $n, n+2, n+4, n+6$, where $n$ is odd. Their product is:

📌 P = n(n+2)(n+4)(n+6)

📌 We seek the largest integer that divides $P$ for all odd $n$. Since $n$ is odd, all terms are odd, so $P$ is odd — 2 does not divide it.

📌 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 $3 \mid P$.

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