If \(n+1\) is even, then \(n+1\) is divisible by 2.
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- 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 \(n+2\) is even, then \(n+2\) is divisible by 2.
- In each case, there is at least one factor of 2.
- For divisibility by 3:
