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📅 April 29, 2026 👤 scraface

Apr 29, 2026

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

📌 Let the three consecutive integers be \(n\), \(n+1\), and \(n+2\). The product is \(n(n+1)(n+2)\). Among any three consecutive integers, at least one is divisible by 2 and at least one is divisible by 3. Therefore, the product is divisible by \(2 \times 3 = 6\).

📌 To see if a larger number must divide the product, consider the following cases:

📌 If \(n\) is even, \(n\) is divisible by 2.

📌 If \(n+1\) is even, then \(n+1\) is divisible by 2.

📌 If \(n+2\) is even, then \(n+2\) is divisible by 2.

📌 In each case, there is at least one factor of 2.

📌 For divisibility by 3:

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