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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:
\text{lcm}(4, 3) = 12
We can confirm this by testing specific cases (like 1, 2, 3 or 2, 3, 4) where the product is divisible by 6 but also by 12 in certain configurations (like 3, 4, 5).
Thus, the largest integer that must divide the product of any three consecutive integers is:
