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 \(2 \times 3 = 6\).

["Understanding Why the Product of Three Consecutive Integers Is Always Divisible by 6", "When we consider three consecutive integers—such as ( n ), ( n+1 ), and ( n+2 )—their product ( n(n+1)(n+2) ) holds a special property: it is always divisible by 6. This fundamental insight is rooted in basic number theory and offers a simple yet powerful mathematical truth.", "### Why Is the Product of Three Consecutive Integers Divisible by 6?", "A number divisible by 6 must be divisible by both 2 and 3. Among any three consecutive integers, two essential conditions are automatically satisfied:", "1. Divisibility by 2 (Evenness):\n In any set of three consecutive numbers, at least one must be even. Why? Because every second number is even, and three consecutive integers cover two even and one odd number (or two odds separated by an even). Therefore, one of ( n ), ( n+1 ), or ( n+2 ) is divisible by 2, ensuring the entire product ( n(n+1)(n+2) ) is even.", "2. Divisibility by 3:\n Likewise, among any three consecutive integers, exactly one must be divisible by 3. This follows from the fact that integers cycle every 3 numbers modulo 3: one of ( n \mod 3 ), ( (n+1) \mod 3 ), or ( (n+2) \mod 3 ) must be 0. Hence, the product includes a multiple of 3.", "### Conclusion: The Product Is Divisible by 6", "Since the product ( n(n+1)(n+2) ) contains at least one factor divisible by 2 and one by 3, and since 2 and 3 are coprime (their greatest common divisor is 1), the product must be divisible by ( 2 \ imes 3 = 6 ).", "This property makes three consecutive integers a fascinating subject in schools and introductory number theory—not just because of its simplicity, but because it reveals how structure in integers guarantees predictable divisibility patterns.", "Whether you're studying algebra, exploring modular arithmetic, or just curious about integer properties, remembering that the product of three consecutive integers is divisible by 6 adds a clear and elegant tool to your mathematical toolkit.", "---", "Key Takeaways:\n- Three consecutive integers include at least one even number ⇒ divisible by 2.\n- They include a multiple of 3 ⇒ divisible by 3.\n- Therefore, ( n(n+1)(n+2) ) is divisible by ( 2 \ imes 3 = 6 ).", "This insight highlights the beauty of basic number theory and is a must-know for students and math enthusiasts alike."]