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Home » \( r^3 = \frac288 \times 34 = 216 \) - Blask

\( r^3 = \frac288 \times 34 = 216 \) - Blask

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 \boxed{14}

📌 What is the largest integer that must divide the product of any three consecutive integers?

📌 Consider three consecutive integers denoted as \( n, n+1, n+2 \).

📌 Among any three consecutive integers, one must be divisible by 2 (since every second integer is even).

📌 Among any three consecutive integers, one must be divisible by 3 (since every third integer is a multiple of 3).

📌 Thus, the product \( n(n+1)(n+2) \) is divisible by both 2 and 3. Therefore, it is divisible by:

📌 Additionally, consider the divisibility by 4. Among any three consecutive integers, at least one is even, and at least one of these even numbers is divisible by 2, making the product divisible by \( 2 \times 2 = 4 \).

📌 Therefore, the product \( n(n+1)(n+2) \) is divisible by the least common multiple of 4 and 3, which is:

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